The winner for the cats would be Persian. I just think Persian is awesome that way. The loser would be Meowth and I think we all agree with that.
For the dogs, that's a bit tougher. The winner would definitly be between Lucario, Arcanine, and Houndoom. I'm favoring Lucario just because I really haven't even used the other two and because Lucario is my favorite. The loser dog would have to be Lillipup even though it's not that fair pairing it up with all the strong pokemon. I do agree that some cat/dog pokemon aren't really for battling.