What is going on is that the tire has a too soft of a side. The sidewall flexes enough that dirt/mud accumulate between the bead of the tire and the rim. Only way to get them clean is to break the bead, clean off the lip of the rim, and reset. If you had a buddy with a tire mounter, it all can be done without re-balancing. You just work one lip at a time. If the tire rotates, it'll be out of balance.
I'd suggest that you look to getting a stiffer sidewall tire with mud-type lugs as a tread pattern.