I've never had to clean a rabbit, but there areseveral other threads about it. Try doing a search onbaths.
I would also suggest that you clean the carrier several times a dayuntil you get him a bigger cage. A good, absorbent litterlike Woody Pet or Carefresh will help, too.
I just want to make sure there are no other underlying problems. Is thecarrier big enough for him to at least sit up and clean hisbackside? If not, I would strongly suggest getting him biggercarrier, if not a cage, immediately. Are all his poopsmushy? Or does he have some mushy poops and some normal hardones? Does he have urine stains? If the carrier istoo small, he won't be able to position himself correctly and will endup peeing on himself. Some health problems can cause this,too. This can cause painful skin irritation called urinescald.