One of the walkthroughs on GameFAQs summed up the Palmer fight perfectly:
This isn't what you would consider to be a hard battle or anything. Not like the Materia Keeper or Gi Nattack, where you actually had to, you know, try. Fighting with Palmer is more like a personal challenge to see how long it takes to make the battlefield explode into a mess of butter and shrapnel. Palmer stands on one side of the screen, wiggling his fat body about in a presumed attempt to move, or perhaps communicate. Occasionally, he pulls out a gun and fires it at you. But since he went and ate all the bullets, thinking them to be Pez, the thing is running on a combination of materia and AAA batteries, and instead mimics the effects of Bolt 2, Fire 2 and Ice 2. But, he is susceptible to the effects of Stop, meaning that if you've got Seal Evil or some other equivalent on hand, he won't even be able to get an attack off. If you don't paralyze him, it's no big deal, but be warned that Palmer will periodically lean over and slap his ass in an attempt to be provocative, but ends up disgusting you more than anything. Kick his ass to high hell, unloading everything you've got onto the chubby guy. The only thing that will not affect him is gravity-based attacks, quite possibly because a body as large as his tends to produce a gravitational force far stronger than anything you could ever conjure up. Anyways, once you've hurt Palmer badly enough, the Tiny Bronco begins to move. Palmer will turn tail and hobble his fat ass along, trying to catch the vehicle that he is attempting to steal. At this point, the game rewards your patience by hitting Palmer with a truck. To add insult to said injury, your team steals his armor (the Edincoat) before leaving him to bleed rich and creamily.