Personally, I've never minded cleaning cats' bottoms or coping with all sorts of unpleasant 'messes' such as vomit etc. (My Siamese vomit at the drop of a hat!) I suppose it's no different to babies' nappies.
But if the kitten's anus isn't kept nice and clean, one risks infection of the anal glands, which is painful and distressing for the cat. I would imagine the mother cat won't want to clean her baby's bottom if it's a case of large amounts of fecal matter. I totally agree about the cat-litter. I've used a product which is recycled newspaper pellets. Very envo-friendly and inert. I do so hope the little one will soon grow out of this problem. Have you consulted a vet who is familiar with tailless cats and their special difficulties? It might be a good idea to get advice now, before problems get worse (touch wood they won't!) Have you a photo of your cats Melissa? I'd love to see them!