AAA is not going to deny towing because you don't carry a spare. None of my cars have spares. They don't come with spare tire because there is no room to put it. However, you are aware that they only tow you a few miles per call and a few calls per year. So if you stuck in a desert 30 miles away from anything... AAA classic membership can't really do much for you even if you get a cell signal to call them. And I don't think you can apply for the 100 miles per tow membership until you have joined the classic membership for a year. Pretty much they don't want you to join for a road trip and not renew your membership next year.

If tire problem is your major concern, maybe just buy a new tire (same size) and throw it in the bed for spare for this road trip. Next time you buy new tires, you only have to get 3.