I think the loser for cats'd be Meowth. It's just weird and weak. Plus in a battle, randomly getting money isn't really gonna help you. Winner would be either Purugly or Luxray. Physical and special respectively...
Dogs? Lucario. I've always thought of it to be smart and plotting, so I think it'd rationally kick butt. Loser would be Lillipup. Even though its evolution is a Giant Lump of Dog Mustache...lol.
But yeah, as for your suggestions, I can't believe I forgot about Lucario. xD Like 90% of cat/dog Pokémon are weak against Lucario so of course he'd win - I can't believe I didn't think of that lmao. Also you have no idea how happy "Giant Lump of Dog Mustache" made me - we definitely need to be friends. xD Does kinda raise the question that are there too many cat/dog Pokémon? I mean, based on the giant dog moustache thing they're obviously running out of ideas D; Not that I can think of many more... Surprised there's no tigers though o3o
I hate to admit that, since I'm a cat-person but generally Pokédogs tend to be tougher :/ Well, of course, I guess this Battle Royal doesn't include the legendaries, otherwise it'd be unfair. So to me the winner would be Arcanine. The final fight would oppose it against Houndoom, the two fire-dogs, but the resistance and the strenght of Arcanine would gain the upper hand, since it would be more a physical battle (using fire-attacks would be a bad strategic choice in view of the opponent).
Then, the first that would be knocked out would be the cute Skitty. And also Snubull. Yes, they would faint both at the same time ^^ I can't see them as real fighters (disregarding their evolutions).