Instead of worrying about keeping it clean, go ahead and mount it on a rim. I have several large trailers that I carry a mounted spare using a come along to raise and lower the tire. Buy one like
this and mount the snatch block to the frame and put a T on the end of the cable. The T needs ears on it that go in the lug holes to keep it centered in the rim so it raises level. Fasten the handle near the rear and while it's not real fast, it will raise and lower the mounted tire. I would put a piece of inner tube on so you can put it around the handle end to keep it clean.
The unmounted tire I carry on the roof needs to be redone as the plastic wrap has deteriorated. Instead of using alum foil I'm going to use some leftover drywall tape and better plastic wrap this time.