On mine, there is a plastic cover that is secured by a bolt that comes up from the bracket, that you would still have to loosen before letting down the spare. And you wouldn't be able to remove the spare from underneath unless you cut that bolt off. If you cut the bolt off and come up with another way to fasten a cover over the tire, maybe it would work. Or just do without a cover, which would not look too sharp.
The other issue I can think of, is there enough clearance under there to actually work with getting the tire out and re-installing it from below, or if so do you want to do that in the rain at night on a gravel road? If I had to move the tire, I would think seriously about the rear bumper mount Jack mentioned. It would be a lot more convenient there.
1999 22L Classsic
2008 Toyota 4Runner V6