Bugs Bunny hasn't said much in 20 years, other than Space Jam there hasn't been a new Warners classic character cartoon since the 60s. "Feets don't fail me now" might be in Space Jam though. Never heard that phrase associated with race ever.
I take it you have never watched a lot of movies from the the 30s and 40s. It was used commonly, usually a black actor would be scared of his/her shadow and when something scary would happen would come out with the expression "feet(not feets) don't fail me now" with his/her eyes bulging out, before running like a scared rabbit(not bugs).
Bill Murray: "Larry Bird isn't white. He's clear."