Updated for formatting, added image links and changed the name Ed to King Bradley.  

I know this strategy is not possible for everyone due to the units needed but I managed to clear this in MP with no continues, two units were controlled by the computer due to our version problems and one of them didn't contribute much so I figured it was worth sharing. I was willing to gem if I was left with just the green dragon but some of the earlier runs were just awful, some due to connection issues.
 

P4 Roxy or Laharl - has to have agi higher than P1. Kills 2 bugs on first turn but do not break down the wall. More bugs will spawn on left side. For Roxy Alchemia burst is not strong enough to kill the bugs. You need to use Quicksilver Bullet while standing on the raised corner. Laharl would need to stand in the same spot and should try to equip a 5* book for the agi boost that way P1 Laharl can have a 3* book/medal ( 8 agi) or a swallow's scarf ( 4 agi).
P1 Laharl - equip gear that increases agi so that you move before the mage. Lvl 75 with scarf (158 agi) is enough. You will be killing 5 bugs and 1 mage leaving you with 1 bug, 3 dragons and King Bradley. Damage shown here will be lower since I took the screenshot in SP. In MP you don't get the use of a leader skill but you can still kill everyone.
P2/P3 - They should both be physical attackers that will help in killing King Bradley and the dragons (2 fire followed by 1 wind dragon).
 

-Roxy's Hollow Point Bullet skill is only 12 hearts and does a lot of damage to King Bradley (approx 1400) as long as it doesn't miss. Magnum Buster shot does slightly more damage but costs 42 jewels. You can also try to null grenade him first so that he won't kill off your units but that's only needed if P2/P3 aren't strong.
-Yomi 85 with 5* hat is good for King Bradley but she will die fast once the fire dragons reach her.
-Vettel is a great. He can tank King Bradley, the dragons and the knight. Move him in front of the wind dragon to keep your other party members safe until it is time to kill him. He's the safest bet on getting to the second cirlce. Downside is that he can't deal much damage but your other 3 units should do that.
-Laharl is needed to help kill the wind dragon (leave it for last). His special takes off over 2700 of his energy without charge up. If you have time to charge up then he can kill the dragon in one shot with blazing knuckle. He can also get from gate 1 to gate 2 in two moves and if he has full health won't die from a hit from the knight.
 

My MP team for this with no continues was Laharl (controlled), Roxy (controlled), Laharl (computer), Roxy (computer). The computer does a good job in controlling Roxy, at least the first few turns. A computer controlled Laharl is terrible. Uses his WA right at the beginning on King Bradley not doing a lot of damage and then dies.
 

I think a good team is Laharl, Vettel, Yomi and Roxy. The first two turns are as per the image links in the P4 and P1 descriptions. Then as long as nobody misses they can kill King Bradley before his second turn. King Bradley moves down to Yomi. Yomi does Supreme Battle Trance on her first turn for approx 2500 damage (regular HB) and attacks him from the back for less chance of evasion. Vettel attacks with a special and Roxy finishes him. Then you kill the bug because you don't want him to go back up and increase the stats for the wind dragon. Kill the fire dragons. Kill the mage when you can. Wind dragon is last. Make sure Laharl has enough jewels to do his special on the wind dragon. Another option is charge up followed by blazing knuckle which will kill him in one shot but the special will get any other enemies that still need to be finished off. You can still get this done if you don't hit King Bradley every time as long as he goes after Vettel or Yomi.
 

This can also be used for single player but will need Laharl or Roxy for slot 1. Slot 4 can be replaced by mercenary. If you use Roxy in slot 1 she can't kill all of the bugs in one shot and has to kill the last bug in that area on her second turn. For no death in SP keep Yomi in the back so she doesn't get hit after attacking King Bradley.