#1,
Do you have or have access to a welder?
If so, have a few questions about the hinged spare carrier you currently have:
Will the upper part reach 90* to the base if the tire is off? Can the base be removed, and turned around 180*?
If yes to both:
My crazy idea is to mount the base on top of the bumper, with the pivot point near the tt back wall. The upper part now is attached to the base, facing to the rear (of course). Weld a couple gussets to keep the upper part 90* to the base, and nice and secure. Spare is now above the bumper somewhat, and a little closer to the tt back wall.
You won't gain a ton by moving it this way, but is an idea to ponder on!!! Otherwise back of tv, under the trailer (not possible per your post), or not sure what else to suggest! Maybe a modded tongue rack that is for a bike rack??? Not sure if you considered that as an option. I know I have seen a rack that sits above the propane tanks for a generator or similar.
Good luck!!!