I don’t think the illustration was supposed to be depicting an actual rabbit, so much as a human woman (presumably with orange skin) wearing bunny-ears. She was wearing bunny ears to show her kinship with Bugs Bunny et al., in that she ate a lot of carrots.
I’d heard that Mel Blanc didn’t like carrots, but that only carrots could make the carrot-munching sound he wanted – so he would crunch on a carrot while saying “What’s up, doc?” and then turn around and spit out the chewed-up carrot parts into a bucket. But I’d never heard that he was actually allergic to carrots.