Did you get Dudley to the vet? He is still small enough that if the vet recommends removing the tail it will grow back. Rambo dropped her tail on us when she was about 6 months old and it has grown back more than I ever expected. I'm anxious to hear what a vet has to say on this subject.
Also, medical tape will help to keep the bandage on and it cleans off nicely in a warm bath.