I took some more measurements today. If I were to build an enclosure to fit within the confines of those foam inserts on either side (just where the tire sits), at 19" x 20" x 8.5", that would only give me 1.3 cf of air space, so the foam will have to go. No real point of keeping it all there anyway if I'm not running a spare.
What I think I'll do is have the enclosure as large as possible, extending somewhat into the those rear corners. That will give me the opportunity to mount the sub off to the side towards one of those corners, probably on the passenger side.
It'll definitely be a little bit of a challenge to shoehorn this sub into the car, but I'll get it done one way or another.
As you can see in the pics below, this Type S 15 is a bit larger than the typical 12's installed in these cars... For obvious reasons.