Great comments from both Brian and Conniesaki.
I probably did have the wrong idea about the Air Hawk, thinking of it more like a typical air mattress or gel pad, and that was fueled by seeing photos of it over-inflated (it seems all the photos, including the store photo in the link posted by Ted, today).
As for the tire size, I have already lowered the bike frame the most I feel comfortable doing with the lowering links. If I were to also use smaller tires, the frame would drop an additional amount and put the bike into a danger zone for hitting things like speed bumps and also overly restrict the lean angle. Although there is one thing it does avoid- and that is suspension travel problems. It could be a useful part of a comprehensive lowering strategy, although to be truly effective, it would need to be both tires, not just the rear. This part of the conversation would be better in the lowering thread instead of here, though. I just wanted to mention that I, like most people, could use some additional padding but that it is typically at odds with leg length (it is the reason MY chosen plan to address seat height had to include a thinner, less comfortable seat, in addition to lowering links and raised boots).