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Free-air subs aren't really meant for an installation like that -- that is, the installation you'll have to do to make it work won't be that different from one that would require a regular, enclosure-mounted sub.
"Normal" subs, we know, use an enclosure for two main reasons -- to contain the sound waves from the rear of the speaker (or manage them so that they complement the sound from the front, as with a ported box), and to provide suspension stiffness by using the enclosed air as a spring.
Free air subs are designed to work with nearly zero air suspension. They usually have stiffer spiders and surrounds than an otherwise similar "normal" sub, allowing them to use a vehicle's entire trunk as an enclosure. That's why you can mount one to the rear shelf of a sedan or coupe. No box is needed, but you need to make the entire trunk airtight so that no other sound leaks back into the passenger compartment.
SO, the idea of mounting a sub in the rear floor, poking into the spare tire, would require that you seal the airspace behind the sub for the system to sound worthwhile. Considering the size of that airspace, you'd be better off with a "normal" sub designed for an enclosure.
HOWEVER, I've been wondering a long, long time if it were possible to fabricate an enclosure that could be OPENED to allow access to the spare tire, which would be contained INSIDE the enclosure. This could avoid taking up hardly any luggage space, possibly only raising the floor an inch or two.
What I've done in the past for a no-brainer sub installation was use a truck-style box laid on its back on the rear floor. I've never really liked the idea of having a bulky "hatchback-style" box. Using a broad, flat box allowed me to use an enclosure with just as much internal volume as a more cubic-shaped box, and still left enough room to carry bulky items & luggage.
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