Would Bugs even mess with Mickey? Bugs Bunny, for the majority, really only goes after people who go after him or threaten harm to others. Very rarely is he the instigator willingly, like the episode where he's the writer making fun of Daffy or the one where he mocks the thought of Elmer Fudd being clever, but the other times when he doesn't know he's the instigator, like the Opera episode, his aggressors don't even attempt to be civil with him the first time and go right to attacking him or destroying his stuff.
And Mickey, especially nowadays, has little intention of starting something with others and would certainly be civil first before aggressive. He and Bugs would be the best of friends and teammates.
Now Donald Duck on the other hand, I can totally see Bugs messing with him, if not because of Donald's cockiness.