Answer:
In my opinion it could be done on the right bike and by a talented seat maker.
First you'd need to measure your current seat to see if the dimensions are close to what you have or would work with your bike since I see no way to alter the shape of the Airhawk (nor would you likely want to).
Your seat maker would need to allow for access to the air valve (small zipper or such) but it's going to be a bit awkward given it's location (rear left corner of pad). You'd likely need to find a rubber or vinyl tube that would slip onto the mouthpiece of the air valve since trying to get to in mounted on the bike would be "interesting" to say the least.
I think you need to ride with it and get it adjusted to you before having the seat made so you'd know what thickness you're dealing with (since it quite variable by the rider). ROHO, the manufacturer, recommends about 1/2" of air but it's really up to you.
Potentially the most difficult part would be not having the cover on the seat restrict the Airhawk since it's designed to move air around as you move or shift in your seat. That means a traditional cover of vinyl or leather could really restrict things, especially if pulled taut to get all the wrinkles out.
Also, the Airhawk has mesh sides on its cover to allow airflow in and around the air chambers to eliminate hot spots. This would be kind of tricky to incorporate into a seat but the right seat maker could probably do it.
Bottom line, I think it could be done but it will be quite the challenge and likely expensive since it will take a really experienced and talented seat maker to get it right. I'd suggest getting one and riding on it for a bit to see if it "does it" for you. If it doesn't you can take advantage of the return policy.
Good luck.