In terms of Pokemon, you really have two threats here- Snivy and Pansage. The two Sewaddle are only really harmful if you have everyone toting around Potions- they’ll Bug Bite the Potions away, healing themselves and denying you your carefully packed curatives, so either wipe them out quickly or just skip having your Warriors keep Potions ready.
Snivy and Pansage are badly neutered by Fire Pokemon, but keep an eye on their Vine Whip attacks- these attacks have a broad, three-tile area that they affect, so packing your Pokemon closely together is a good way to invite an excess of damage, particularly on your and Oichi’s Normal-type Pokemon. Since the Fire-types you’ve gained will generally not care about these Grass attacks, it’s to your favor to keep your Jigglypuff and Eevee on two of the banners while you squabble with Motonari over the third one.
Before anyone gets to take a turn, Motonari inquires as to what you believe is the key for winning battles. He suggests that you are thinking only of strength, and points out that power alone does not secure a win. This is followed up by the computer explaining how banners work as a method of claiming territory. These banners are the win condition for the fight, so you don’t technically have to knock out all the enemy Pokemon to succeed. While this is true, you gain more from knockouts, as they give you a chance to recruit a Warrior and furthermore, they improve the Link ratings of your Warriors’ Pokemon at the end of the battle.
At the end of your first turn, you get a rude surprise- the gap in the middle of the map grows a vine bridge! This bridge only lasts a couple of turns at a time, and there’s nothing on the far side of it that you want- only things you want to keep away. Parking one of your Fire Pokemon at the base of it, on the stairs, can provide you with some considerable advantage.
After the bridge grows, one of the opposing Warriors will have his Pokemon step out onto it, prompting Oichi to point out the banners, suggest splitting up, and re-state her obedience to you. Presumably, she considers this encouraging.