if you are only buying AAA for the hotels discount do not do it, i have one but hotel discounts are not the reason I got it, thats just a side benefit.
Although not the most ethical of ways, yeah yeah I hear the ethics police out here, you could apply for a membership, get your card, which may be good for two years, it will have your name on it, then cancel.. I "think" you can get a refund. its no different than those booking the AAA rate on Marriott, Hyatt and SPG who don't have a card either. I have been asked for my AAA card once in my life and of course i didn't have it on me LOL and they still let me have it when I said, let me go online and print out my account information from AAA
you will have a number to use when booking etc.