I'm pretty sure all bronze medal pets fight. Just pets worth bronze medals do the same damage as you, so it seems they don't do anything.
If you want a pet that'll do more damage than you do, then you'll probably need to get a 4+ gold pet. uvu'' Anything lower prices doesn't really do much. The only pets (that I know of) that don't attack are worth 2 and 3 gold, and those are praise pets.