This trial took... a lot of time... to work through. The AI does some really complex, gimmicky stuff, and all of Tilith's allies have more complex AI than "BB every turn, SBB every 4th turn, UBB at a threshold" that summoned units normally have in trials.

Anyways, I'm going to filter out the more complicated stuff and just have the stats and AI overview in this main post

All damage estimates assume the player has 0 DEF and no mitigation, so usually the real damage will be significantly less. Mitigation applies to most damage used in the trial (some attacks reduce its effects a little, Phase 3 Tilith has one buff wipe), while DEF doesn't apply to fixed damage or HP% damage (or while Lugina's DEF ignore is active).

Phase 1 - 6* Tilith, 7* Lugina, 7* Paris

6* Tilith (Phase 1)

Stats and Resistances

Actions per turn: 3~4
HP: 4000000 ATK: 5000 DEF: 3000

Ailment Resistances:

• 100% to all core ailments
• No resist to stat downs

Other Resistances:

• 30%/50% (Base/Buffed) BC Drop Rate Resistance
  • Base BC drop rate reduced to 24.5%
  • Value of all bonuses halved
• 30%/50% HC Drop Rate Resistance
  • Base HC drop rate reduced to 7%
  • Value of all bonuses halved
• 10%/80% Crit Chance Resistance
  • Base crit chance reduced to 9%
  • Value of all bonuses reduced by 80% (multiplied by 0.2)
• 50%/100% Crit Damage Resistance
  • Base crit damage reduced to 25%
  • Crit Damage bonuses do nothing
• 50%/100% Elemental Weakness Resistance
  • Total immunity to base elemental bonus (additive effect for whatever reason)
    • Given how this is used I'm starting to think Alim made it a multiplier, though, in which case instead reduced to 25%
  • -100% to total value of Elemental Weakness Damage sources (or immunity?)
  • Can't reduce damage below 100%
• 30%/80% Spark Damage Resistance
  • Base reduced to 35% damage increase
  • Value of all bonuses reduced by 80% (multiplied by 0.2)

Other Passives:

• 20% bonus to Tilith's Elemental Weakness Damage
  • Total multiplier vs Dark units is 170% instead of 150%

AI Overview

• 500k heal every 4 turns if player constantly uses 3 or more BB
  • Using 2 BB on any turn resets the timer
  • Warning that next turn will trigger: "『白のヘミスフィア』が輝きだした"
• 150-200k regen applied if player uses a UBB
  • Only once
  • If this is triggered, Tilith will wipe your squad at the start of Phase 2
• Thresholds:
  • 80% HP: Summon Lugina and Paris
  • 60% HP: 160% HP damage on highest HP unit
  • 40% HP: 160% HP damage on highest ATK unit
  • 20% HP: 160% HP damage on highest DEF unit and on lowest HP unit
• Timers:
  • 3 turn timer: Multi-elemental AoE
    • ~10k damage with no buffs
  • 5 turn timer: AoE 100k heal
    • ~8k damage with no buffs
• Normal attack priority:
  • Units under 75% HP: 25% chance
  • Dark units: 20% chance
  • Else, random

Lugina

Stats and Resistances

Actions per turn: 2~3
HP: 4000000 ATK: 9500 DEF: 2000

Ailment Resistances:

• 80% resistance to Injury
• 75% resistance to Sickness
• Immunity to all other core ailments
• No resist to stat downs

Other Resistances:

• 35%/60% BC Drop Rate Resistance
  • Base BC drop rate reduced to 22.75%
  • Value of all bonuses reduced by 60% (multiplied by 0.4)
• 60%/50% HC Drop Rate Resistance
  • Base HC drop rate reduced to 4%
  • Value of all bonuses is halved
• 10%/80% Crit Chance Resistance
  • Base crit chance reduced to 9%
  • Value of all bonuses reduced by 80% (multiplied by 0.2)
• 10%/90% Crit Damage Resistance
  • Base crit damage reduced to 45%
  • Value of all bonuses reduced by 80% (multiplied by 0.1)
• 10%/80% Elemental Weakness Resistance
  • Base Elemental Damage bonus reduced to 40% (45%?)
  • -80% to total value of Elemental Weakness Damage sources (or x0.2?)
  • Can't reduce damage below 100%
• 30%/80% Spark Damage Resistance
  • Base reduced to 35% damage increase
  • Value of all bonuses reduced by 80% (multiplied by 0.2)

Other Passives:

• 25% elemental weakness damage dealt to thunder units
  • 175% damage multiplier instead of 150% when Lugina attacks a thunder unit

AI Overview

• If Tilith passes 10% HP in phase 1, or 99% HP in phase 2, use Overdrive ("オーバードライブ")
  • Turn 1: 50% ATK/DEF, 3 turns; 70% mitigation, 2 turns; -70% curse resistance, 2 turns
  • Turn 3: UBB (Single target attack, ~72k damage)
    • Can be prevented by cursing Lugina on previous turn
  • Turn 4: SBB (AoE, ~15k damage, 2 turn DEF Ignore to Tilith squad)
    • Only happens if UBB prevented via curse
  • Alternately prevent Lugina's Overdrive entirely by killing him before taking Tilith under 10% HP in phase 1
• 80% Threshold: SBB (See above)
• 50% Threshold: 50k fixed damage attack on highest ATK unit
• 30% Threshold: SBB (See above)
• After passing the 50% threshold, the following timers are active:
  • 4 turn timer: Self buffs for 1 hit count, 50% ATK/DEF
  • 6 turn timer: SBB (See above)
• Permanently active 3 turn timer:
  • Single Target attack for 160% HP% damage
    • 70% chance to specifically target a Thunder unit if any are present
    • HP% damage isn't affected by elemental weakness, though
• Normal attack priorities:
  • Units over 50% HP: 25% chance
  • Highest ATK unit: 50% chance
  • Else, random

Paris

Stats and Resistances

Actions per turn: 2~3
HP: 3400000 ATK: 8000 DEF: 5000

Ailment Resistances:

• 60% resistance to Sick
• 50% resistance to Weaken
• Immunity to all other core ailments
• No resist to stat downs

Other Resistances:

• 80%/100% BC Drop Rate Base Resistance
  • Base BC drop rate reduced to 7%
  • BC Drop Rate bonuses do nothing
• 80%/100% HC Drop Rate Base Resistance
  • Base HC drop rate reduced to 2%=
  • HC Drop Rate bonuses do nothing
• 10%/80% Crit Chance Base Resistance
  • Bae crit chance reduced to 9%
  • Value of all bonuses reduced by 80% (multiplied by 0.2)
• 10%/90% Crit Damage Base Resistance
  • Base crit damage reduced to 45%
  • Value of all bonuses reduced by 80% (multiplied by 0.1)
• 10%/80% Elemental Weakness Base Resistance
  • Base Elemental Damage bonus reduced to 40% (45%?)
  • -80% to total value of Elemental Weakness Damage sources (x0.2?)
  • Can't reduce damage below 100%
• 30%/80% Spark Damage Base Resistance
  • Base reduced to 35% damage increase
  • Value of all bonuses reduced by 80% (multiplied by 0.2)

AI Overview

• When Paris is below 20% HP, she'll use Overdrive ("オーバードライブ")
  • Turn 1: Self-buffs 50% ATK/DEF for 3 turns, 70% mitigation for 2 turns, then ends turn
  • Turn 2: Normal attacks only
  • Turn 3: Uses UBB or BB
    • UBB: Massive AoE (~44k damage), 200% ATK/DEF and 99-100k HP regen to Tilith's squad for 3 turns, 100% mitigation to Tilith's squad for 1 turn
    • BB: Small AoE (~14k damage), 50-90k heal and 1 turn ailment immunity to Tilith's squad
    • BB is used if the player uses BB with all units on the last turn before UBB fires. This prevents UBB from happening at all.
• 90% HP threshold: Single Target Nuke (15k fixed damage) on unit with highest DEF
• 60% HP threshold: Small AoE (~12k damage) and chance of 5 turn spark vulnerability applied to player squad (50% chance, 25% spark damage received)
  • Attack is repeated on a 5 turn timer
    • The timer actually runs from the first turn of the fight, but can only trigger after 60% HP
    • Time version and threshold version cannot trigger on same turn, allows for preventing one use of the skill
• 40% HP threshold: Single Target Nuke (15k fixed damage) on unit with lowest DEF
• 4 turn timer: AoE HP% damage attack (60-80%)
  • Only after the 60% threshold has been passed
  • Cannot happen on the same turn as the 5 turn timer above
• 3 turn timer: BB
  • Small AoE (~10k damage), 50-90k heal and 1 turn ailment immunity to Tilith's squad
• Normal attack priorities:
  • Water units: 20% chance
  • Units over 50% BB gauge: 20% chance
  • Else, random

Phase 2: 6* Tilith, 7* Seria, 7* Karl

Tilith 2

Stats and Resistances

Actions per turn: 3~4
HP: 4500000 ATK: 8000 DEF: 2000

Ailment Resistances:

• 100% to all core ailments
• No resist to stat downs

Other Resistances:

• 30%/50% BC Drop Rate Resistance (Base/Buffs)
  • Base reduced to 24.5%
  • Value of all bonuses halved
• 30%/50% HC Drop Rate Resistance (Base/Buffs)
  • Base reduced to 7%
  • Value of all bonuses halved
• 20%/80% Crit Chance Resistance (Base/Buffs)
  • Base reduced to 8%
  • Value of all bonuses reduced by 80% (multiplied by 0.2)
• 50%/100% Crit Damage Resistance (Base/Buffs)
  • Base reduced to 25%
  • Crit Damage bonuses do nothing
• 50%/100% Elemental Weakness Resistance (Base/Buffs)
  • Total immunity to base elemental bonus (additive effect for whatever reason)
    • Given how this is used I'm starting to think Alim made it a multiplier, though, in which case instead reduced to 25%
  • -100% to total value of Elemental Weakness Damage sources (or immunity?)
  • Can't reduce damage below 100%
• 30%/80% Spark Damage Resistance (Base/Buffs)
  • Base reduced to 35% damage increase
  • Value of all bonuses reduced by 80% (multiplied by 0.2)

Other Passives:

• 20% bonus to Tilith's Elemental Weakness Damage
  • Total multiplier vs Dark units is 170% instead of 150%

AI Overview

• If a UBB is used before Tilith passes 50% HP, Tilith will wipe the player squad and give her squad 150-250k HP regen
  • Additionally, she'll use this attack if you used a UBB during phase 1 on any turn except the turn you finished it off on
• On turn 1, Tilith will debuff the player squad's BB modifiers
  • -200% to BB
  • -300% to SBB
  • -800% to UBB
  • Duration unknown, looks to be 20 turns?
• Tilith follows different BB usage rules for this phase:
  • Turns 1-3: If the player uses BB with 5 or more units, Tilith will heal her squad for 500k HP and restart the timer
  • Turn 4: Tilith will say "『光のヘミスフィア』が輝きだした" and give herself HP regen.
  • Player must then use BB with all living units or face the following effects:
    • -50% ATK, -80% DEF, 2 turns
    • -70% BB fill rate, 1 turn
    • 90% BB drain
• 80% HP threshold
  • Tilith summons Seria and Karl
• 60% HP threshold
  • Single Target nuke (160% HP damage) on Highest HP unit
• 20% HP threshold
  • Single Target nuke (160% HP damage) on Lowest HP unit
  • Additional Single Target nuke (160% HP damage) on Highest ATK unit
• 3 turn timer: Multi-elemental AoE (~12.8k damage)
  • Additionally this AoE can be used at a 20% chance instead of normal attacks if Tilith is alone
• 5 turn timer: Light AoE (~13.6k damage) 99-100k heal for Tilith's squad
  • Timers can overlap
• Normal attack priorities:
  • Units with HP under 75%: 25% chance
  • Dark units: 30% chance
  • Else, random

Seria

Stats and Resistances

Actions per turn: 4
HP: 3700000 ATK: 10000 DEF: 3000

Ailment Resistances:

• 80% resistance to Injury and Weaken
• Immunity to all other core ailments
• No resist to stat downs

Other Resistances:

• 30%/50% BC Drop Rate Resistance
  • Base reduced to 24.5%
  • Value of all bonuses halved
• 60%/100% HC Drop Rate Resistance
  • Base Reduced to 4%
  • HC Drop bonuses do nothing
• 10%/80% Crit Chance Resistance
  • Base reduced to 9%
  • Value of all bonuses reduced by 80% (multiplied by 0.2)
• 10%/90% Crit Damage Base Resistance
  • Base reduced to 45%
  • Value of all bonuses reduced by 80% (multiplied by 0.1)
• 10%/80% Elemental Weakness Resistance
  • Base Elemental Damage bonus reduced to 40% (45%?)
  • -80% to total value of Elemental Weakness Damage sources (x0.2?)
  • Can't reduce damage below 100%
• 30%/80% Spark Damage Resistance
  • Base Reduced to 35% damage increase
  • Value of all bonuses reduced by 80% (multiplied by 0.2)

AI Overview

• If Tilith's health is below 99% in phase 3 or below 20% HP in phase 2, Seria will activate her Overdrive sequence ("オーバードライブ")
  • Turn 1: 100% mitigation, 50% ATK/DEF, turn ends
  • Turn 2: Normal attacks only
  • Turn 3: UBB (AoE ~30k damage base, 2 turn DoT for ~11k, -80% ATK for 2 turns)
• 50% threshold: buffs hit count ( 2) and HP regen (10-30k) until end of fight and changes normal attack AI
  • Only delayed by Overdrive sequence, not prevented
• 80% and 60% thresholds: Heavy Single Target attack (random target)
  • ~19k damage base
  • Only delayed by Overdrive sequence, not prevented
• 3 turn timer: Same attack as 80% and 60% thresholds (random target)
• Before passing the 50% threshold, Seria appears to skip every 3rd turn (cannot prevent UBB/overdrive)
  • Honestly I'm not sure what this piece of AI is intended for, there's just a random turn end script in there.
• After passing 50% threshold, 10 turn timer: Medium AoE and 100% ATK down
  • ~16k damage base
  • Note: Skipped during Overdrive sequence
• 10% chance instead of any normal attack: BB
  • Approx 14.7k damage, AoE
• Normal attack priorities:
  • Above 50% HP:
    • Units over 50% BB gauge: 25% chance
    • Lowest HP unit: 20% chance
    • Else, Random
  • Below 50% HP:
    • Earth units: 15% chance
    • Lowest ATK unit: 25% chance
    • Lowest HP unit: 50% chance
    • Else, random

Karl

Stats and Resistances

Actions per turn: 3
HP: 4100000 ATK: 8500 DEF: 4000

Ailment Resistances:

• 50% resistance to Weaken
• Immunity to all other core ailments
• No resist to stat downs

Other Resistances:

• 30%/80% BC Drop Rate Resistance
  • Base reduced to 24.5%
  • Value of all bonuses reduced by 80% (multiplied by 0.2)
• 50%/50% HC Drop Rate Resistance
  • Base Reduced to 5%
  • Value of all bonuses halved
• 25%/75% Crit Chance Resistance
  • Base reduced to 7.5%
  • Value of all bonuses reduced by 75% (multiplied by 0.25)
• 10%/90% Crit Damage Resistance
  • Base reduced to 45%
  • Value of all bonuses reduced by 80% (multiplied by 0.1)
• 10%/70% Elemental Weakness Resistance
  • Base Elemental Damage bonus reduced to 40% (45%?)
  • -70% to total value of Elemental Weakness Damage sources (x0.3?)
  • Can't reduce damage below 100%
• 25%/75% Spark Damage Base Resistance
  • Base reduced to 37.5% damage increase
  • Value of all bonuses reduced by 75% (multiplied by 0.25)

AI Overview

• 20% HP Threshold: activate Overdrive and end turn ("オーバードライブ")
  • Turn 1: 50% ATK/DEF for 3 turns, 100% mitigation for 1 turn, ends turn
  • Turn 2: Normal attacks only
  • Turn 3: UBB and end turn
    • ~34k damage
    • UBB will be replaced with a regular BB (~16k damage) if player uses BB with all units on the turn before UBB fires
    • If UBB is prevented, turn will not end and timers can trigger
• 80%, 60%, 30% thresholds: Heavy Light element single target attack (random target) that inflicts DoT
  • ~18k damage, 20% reduced mitigation effectiveness vs this attack
  • ~10k DoT, 2 turns
  • Repeated on an 8 turn timer while below 50% HP (timer runs from start of phase 1)
• 50% threshold: Self-buffs 50% mitigation, 10-30k HP regen and 30% ATK for 3 turns. Self-debuffs -80% Injury resistance for 3 turns
  • Repeated on a 6 turn timer after that (timer runs from start of phase 1)
• 23.5% chance to use a heavy single target BB instead of any normal attack.
  • Can be either (60% chance):
    • ~18k damage (Earth element), 20% reduced mitigation effectiveness vs this attack
    • 50% chance to inflict Injury and Weaken
  • Or (40% chance):
    • ~18k damage (Thunder element), 20% reduced mitigation effectiveness vs this attack
    • 60% chance to inflict Paralysis
• Karl's normal attacks have no targeting priorities

Phase 3 - 7* Tilith

7* Tilith

Stats and Resistances

Actions per turn: 4
HP: 4000000 ATK: 15000 DEF: 3000

Ailment Resistances:

• 100% to all core ailments
• No resist to stat downs

Other Resistances:

• 50%/80% BC Drop Rate Resistance
  • Base reduced to 17.5%
  • Value of all bonuses reduced by 80% (multiplied by 0.2)
• 20%/90% HC Drop Rate Resistance
  • Base reduced to 8%
  • Value of all bonuses reduced by 90% (multiplied by 0.1)
• 50%/80% Crit Chance Resistance
  • Base reduced to 5%
  • Value of all bonuses reduced by 80% (multiplied by 0.2)
• 50%/100% Crit Damage Base Resistance
  • Base reduced to 25%
  • Crit Damage bonuses do nothing
• 50%/100% Elemental Weakness Resistance
  • Total immunity to base elemental bonus (or just reduced to 25%?)
  • -100% to total value of Elemental Weakness Damage sources (or total immunity?)
  • Can't reduce damage below 100%
• 25%/75 Spark Damage Resistance
  • Base reduced to 37.5% damage increase
  • Value of all bonuses reduced by 75% (multiplied by 0.25)

Other Passives:

• 25% bonus to Tilith's Elemental Weakness Damage
  • Total multiplier vs Dark units is 175% instead of 150%
  • Only applies while Tilith is Light element

AI Overview

• All timers in this phase are running from the start of the very first phase
• On a 5 turn timer, Tilith will use two single target nukes, then BB drain your entire squad
  • ~18k damage ( 1800-2100 depending on Tilith's current element)
  • Mitigation is slightly (~17.6%) less effective than normal
  • These nukes will target your two highest HP units
• On a 4 turn timer, Tilith will use her BB
  • AoE ~16.5k damage ( 1500~1750 depending on element) 99-100k heal for Tilith's squad
• If Tilith is alone, she'll use her multi-elemental AoE on every turn except BB turns
  • AoE ~19.8k damage ( 1650-1925 depending on element)
  • Mitigation is slightly (9.1%) less effective than normal
  • She will use that AoE an extra time on all odd-numbered turns, whether she's alone or not
• While above 50% HP, Tilith will change elements and receive various buffs/debuffs every 2nd turn:
  • Light element: Used if at least 3 of Tilith's allies are still alive somehow. 10% ATK, gives Tilith an Angel Idol
  • Fire element: 20% chance. 10% ATK, Tilith becomes vulnerable to Injury
  • Water element: 20% chance. 10% BB ATK, Tilith becomes vulnerable to Sickness
  • Earth element: 20% chance. 15% ATK/DEF, Tilith becomes vulnerable to Injury
  • Thunder element: 20% chance. 25% chance of damage reflect, Tilith becomes vulnerable to Weaken
• At 50% HP, Tilith will use a skill that does nothing (just dialogue), then change her element back to Light and gain an angel idol buff. The angel idol will be recast every 8 turns.
• Once that's happened, a new rule is in effect: For every 12 BB the player squad uses, Tilith will use a single target nuke.
  • 60% HP% damage and buff-wipes the target
    • 25% chance of targeting lowest ATK unit
    • 25% chance of targeting lowest DEF unit
    • 50% chance of targeting lowest HP unit
• At 20% HP, Tilith will gain 100% mitigation for 3 turns and decrease player's BB fill rate increasingly
  • 1st warning skill: "『虹のヘミスフィア』が光を集めていく…"
  • 2nd warning skill: "『虹のヘミスフィア』が輝き出す…"
  • Final warning skill: "『虹のヘミスフィア』が大いなる光を宿す！"
  • After last turn, Tilith will use her UBB, "奇跡のゲート" (30k damage, 800k heal for Tilith)
    • Can be prevented by using 20 BB total during charging sequence
    • Using a UBB counts as 3 BB uses during this counter specifically
    • The previous BB rule is suspended during this sequence, but restarts once it ends
  • If the sequence is broken, Tilith will fully heal your squad including reviving any dead units and filling all BB/OD gauges
    • Every 3 turns after this she'll refill your BB/OD gauges
  • If the sequence is not broken, it will restart when Tilith reaches 20% HP again
    • The BB usage counter will not be reset, so you can work towards 20 BB uses over multiple Gate sequences
• Normal attack priorities
  • Units under 75% HP: 25% chance
  • Dark units: 20% chance
  • Else, random