It sounds like brush mark removal designs may need a lower center weight to aid stability. It won't fix it, but it could probably help reduce that much sliding around.
Should place some 1" cleats across the roller to bite into the snow and force rotation so it doesn't pile snow in front of it. Something like 1"x1" L bar. A length bolted every 6 inches or so. Alternatively, could use some sheet metal to build a skid plate so it forces the roller to go over the snow instead of pulling through it.
Making the trail trimming trailer into a snow roller, then towing it down some snowmobile trails to pack the snow and trim branches.