Look who pulled the shiniest disco ball in town, kupo! I hear you don't even know what it does, so I am here to help you, kupo!

(credits to CareerSMN for compiling this info over here)

Orlandeau OSB Multiplier

15x with 3.75s cast time

Orlandeau BSB Multiplier^{credits to Zurai001}

BSB entry: 10x1.0 random hit holy/dark physical attacks with 100% Instant KO chance, 3.75 second cast time (10.0x)
Justice Sword: 4x1.0 single target holy/non physical attacks with Attack Defense -20% to the target for 15 seconds, 3.3 second cast time (4.0x)
Darkness Sword: 2x1.9 potency single target dark/non physical attacks with 20% HP leech each, 3.3 second cast time (3.8x)

Best setups to use on TGC (OSB only)

Lifesiphon any skill of your choice is enough:

• Gaia Cross: damage Draw Fire in one.
• Powerchain: if you ever want to instant cast that first OSB - you are unlikely to use this after the first cast anyway.
• Saint's Cross/Full Charge/Twinstrikes: 1/3 CT powerful DPS abilities.
• Hailstorm/Dark Bargain: in case you ever want to stack more ATK
• Lifesiphon: a second one because why not?

RM choices:

• Damage up: For much thicker bosses, this helps you hit the damage cap easier. Orlandeau is generally near the ATK softcap anyway, between his 6* Holy sword and another Holy ATK piece of armor (i.e. Platinum Shield).
• ATK up (any variant): If you're not running an ATK buff, this might be the best choice.
• Ace Striker/Battleforged/Lionheart: If you think you're already going to cap 99999 damage without the help of a damage RM, then these are the best choice. Lionheart might work better on a physical heavy boss you expect to Draw Fire/Magic Lure a lot from.

Best setups to use on TGC (BSB only)

You're probably not going to be too well off here, but the likeliest best combination is to use Powerchain into any of the commands (most of the time, it'll be C1).

Another alternative if you can spare it is to RW TGC OSB on him instead before you use the BSB, for the Thundergod buff. Oftentimes, he will still do great damage regardless, so this is never really a bad choice.

The last alternative is to equip your own TGC with his Lv99 RM, then have someone Entrust a bar to him, so you can start off with the BSB right away.

(credits to akanzaki for compiling this info over here)

TGC OSB/BSB Math

It seems that every time I see a topic regarding TGC’s OSB & BSB, someone is always claiming that BSB is totally useless as straight OSB spam with lifesiphons is better damage than using OSB BSB. The issue is that they are usually using multipliers of the two setups to get to a damage number, without accounting for comparable turn times, or the buffs from the BSB (including buffs from burst mode), or the debuff from Burst command 1. I did the math back when attack soft cap was 700, assuming enemy DEF=4000, you start at the attack soft cap, neutral element, and TGC gets hit once per 2 turns.

Here is the result

Not only is OSB BSB better for DPS, you get an additional ATK/DEF -20% debuff on the enemy, and while the straight OSB option is totally tapped out after these 20 lifesiphons, you still have gas in the tank for another 27 turns on the BSB option. If your starting ATK is lower (which, without a 8* synergy weapon, it will be), that’s more dps for the BSB option due to the ATK buff. If you are facing a holy-weak boss, the BSB loadout is MUCH better since OSB will run into 99999 damage cap issue. And now that attack softcap is 805, that’s even more points for the OSB BSB option. So whoever out there is spreading misinformation that OSB spam > OSB BSB in all cases, just stop. It’s wrong, and misleading for people who haven't looked into it.

Note: As pointed out in the comments here by Road-- , a TGC using DMG 30% RM that does not hit the OSB damage cap will slightly outdps the OSB BSB option. There are many benefits of the BSB other than damage, however, and this OSB hits the damage cap fairly easily, especially against holy-weak bosses. In this case, you are much better off going w/ the OSB BSB combo.