Win condition: Defeat commander
The goal here is to defeat the commander, but you're not going to get close to Barth without essentially wiping out his whole army anyway. The green allied units can fend for themselves mostly, though Leon is a bit fragile, so you may want to head north at a faster clip if saving him is important to you. Kamui and especially Valbar, however, will be just fine. You can recruit each allied unit after this battle concludes, so make sure anyone you're interested in survives.
Proceed from your ship over the left plank, leading with strong units and letting Mages assault from behind – you know the drill by now. This map is substantially less difficult than the prior one, as long as you're careful. You can use illusory soldiers via Invoke in a pinch, but they aren't necessary here, and the loss of precious XP when they defeat units for you isn't really worth it in this case.
Once you've cleared the map, head toward Barth. You can have Mae or Boey attack him from behind the southern fortress wall using Thunder, and send Celica and Saber in for some melee. Odds are Valbar will absorb a lot of the hits before you have to, by which point you should be able to wear down Barth's health to complete the map. If you want to recruit Kamui, be careful he isn't killed here.