I use a wooden branch as well, and this is what I do to clean it. First I use that blue reptile cleaner with the long name (goes to check) chlorihexidine. I soak the poop stains in that for about 5 minutes, and then I take it outside and shoot it with the power washer we use for our house and sidewalk. Looks as good as new by the time we're done, and no smell remains either.
(I know how you feel. Until just recently, that is the ONLY place Toothless will go.)