When clash of powers from different worlds are hypothesized about, you have to take into consideration that it probably mostly comes down to the universe itself, and not the character.
Bugs Bunny in Superman's world would be defeated because the setting is more "realistic", so bunny would be a decent superhero at best, but not top-tier.
Superman in Bugs Bunny world would be depicted as the white chad that lifts weights and is good at sports, but ultimately succumbing in a comical way if confrontation were to arise. I can see the episode already: bunny tries all kinds of tricks against him, even hits him with an oversized mallet, which crumbles like ice on top of Superman's head, but then gets beaten up and sent to the hospital. Eventually, bunny recovers and learns the weakness of Superman, then orders Kryptonite from a snake-oil merchant, and makes superman eat it, which turns him normal. Then bunny gets his boxing gloves out, and beats the shit out of him, sending him into a state of unrecoverable coma.
Th-th-th-that's all folks!