mrdabog, each of our Odys (2002 & 2003 EX's) have a full-size spare mounted in back, behind the third row seat. We tow an 18-foot fibreglass bass boat on a galvanized steel trailer, so the towed weight is a little heavy...like you, I've realized that the reduced-size spare is not our first choice for addressing a roadside flat when trailering.
I'm using the spare tire hardware kit to secure these full-size spares. That hardware kit is normally stowed away, and is normally only used when you need to mount the flat tire inside the car, after installing the reduced-size roadside spare.
My full-sized spare on my 2003 is a previously-damaged OEM alloy wheel I got from a neighbor for free. The people at Wheel Connection in OKC fixed it to like-new for $60.
My wife's 2002 full-sized spare is on an OEM steel wheel (like the one's on the Odyssey LX version). I am cleaning up another damaged OEM alloy wheel to send to Wheel Connection later this week, and if that works out, I can send you my OEM steel wheel (15"x6"JJ). soareyes sent it to me for shipping only, and I can do the same for you....
...again, if that OEM alloy I've got in my garage can be repaired, then the OEM full-sized steel wheel is yours for the asking. It is in perfect shape.
OF