Love the implication here that the one ring would have little to no effect on Bugs

To be fair, it’s canonically established in Lord of the Rings that Tom Bombadil, an inexplicable magical trickster, is unaffected by the ring, and the only reason they don’t give the job to him is because Tom Bombadil is a silly little man who’s easily distracted and just wants to spend time with his hot wife.

Bugs Bunny, on the other hand, loves nothing more than fucking over self-important dickheads, and is also an inexplicable magical trickster, so he would in fact be perfect for this mission.

The One Ring may not tempt Bugs, but he’d have other problems with the mission: he’d get lost halfway there (”I knew I should’ve made a left turn at Albuquerque”) and get distracted enough to hand the One RIng to Elmer Fudd or Yosemite Sam as a prank, only for it to be stolen by Daffy Duck, leading to an ever-increasing number of characters on an increasingly-destructive chase across Middle Earth as everyone keeps stealing it from each other, (Bugs would definitely pull the “evil volcano inspector” gag to get into Mordor, and he’d then immediately turn around and pose as a customs agent stopping whoever currently has the ring at the border and relieving them of it as “contraband”) culminating in an all-out brawl at Mount Doom.  Bugs manages to reclaim the ring one last time as everyone else is busy fighting each other, only for Daffy to come out of nowhere and grab it out of his hands.  Laughing maniacally, Daffy doesn’t realize that his victory dance has taken him right off the edge off a cliff - until Bugs points it out, at which point gravity reasserts itself, and Daffy and the ring both plunge to the fiery depths below

Daffy would just be crawling out of the volcano, feathers completely gone and he’s brunt to a crisp while Bugs is eating a carrot and just says:

“Got a good tan, doc?”
“YOU’RE DESPICABLE!”