I would never advise that you give a bunny a bath unless the bunny has soiled himself and he needs a butt bath......

Bunnies are very clean animals, if you watch them closely you will notice that all they do is eat, groom, play once in a while and nap. A bath can traumatize a bunny, he may fight you, and in fighting to get away from something unnatural, like a bath, may harm himself by breaking a leg his back or neck. It just is not worth the risk, especially for something as minor as a carrot stain.... With my bunnies, carrot stains disappear usually within a couple of hours due to their constant grooming.