Does the stuff that's needing to be cleaned off actually impair the bow's performance or is it just a looks thing? If it's just for appearance, why not dye the hair black? Then you won't have to worry about it again.

Heck, I've seen teenagers at festivals using bows with the hair dyed any color you can imagine, sometimes multiple colors on the same bow. Some of them even had the hair on their heads died to match their bows.