Watch enough Looney Tunes and you'll know there's never going to be an "all-out brawl." They'd have one or two orcs, max. Termite Terrace was churning these things out at an amazing rate, which meant there was little time for multi-character interactions. Watch the classic Chuck Jones, Friz Freleng, Frank Tashlin, etc shorts and notice how many characters are on screen at once. It's usually only one, even when in conversation. Also the number of establishing and background panning shots eat up a lot of footage that would otherwise be animated. So, yes, Bugs would hand Mordor their ass, but the expensive spectacle simply would not happen within the known animation restraints of the Looney Tunes/Merrie Melodies oeuvre.
After Daffy and the ring plunge into the depths, 30 seconds later Daffy comes from off screen, smoking with most of his feathers burnt off, the ring no where in sight. He stops in front of Bugs, points at him, and mutters, "you're despicable," then stalks off the opposite side of the screen.
Thank you for coming to my FUDD talk.
(I mean, don't, but for reals, this is the crossover we deserve).