I took my entire tire off my TT bumper and carry it in the back of my pick-up. Reason? 1st, I've NEVER had to use the spare. 2nd, my camper came with a mounted 1 1/2 inch hitch, so purchased a carrier for the back to carry extra hoses (sewer and water). No room for the tire mount now. 3rd, I lost my tire cover on my last trailer, blew off in the wind or stolen. Who knows? But never replaced it. If it caused undue stress on the spare tire, I never knew it, since it never was taken off the bumper in 8 years!
My opinion? Cover the tire. It looks nicer. But get a cover with the open bottom like you use on the tires on the ground, and tie up the two eyelets real snug.