You could top the litter with hay or a grate. Either one will keep bunny from getting soiled even if he lays in the litter box. The hay can be refreshed twice per day if you choose that option. That is what I do - the hay rather than the grate.
I assume you are doing just butt baths - not the whole rabbit. (Whole baths are warned against.) If doing the above options don't work, then my guess would be that something is off on his diet that is causing mushy poos. Normal poos shouldn't stick to fur, especially if one of the above steps is taken.