I've not checked in to a Disney resort with a AAA rate yet since I'm a new member (as of July), and my trip isn't until Oct. However, I've booked 3 different AAA reservations over the past month, and only 1 of the 3 CMs I spoke to while booking these reservations asked me for my membership number and expiration date. So it depends on who books your stay. So, as to whether or not you'll have to show a AAA card at check in, I'm sure it will just as likely depend on the CM who checks you in at your resort.
SO--I'd say it's better than a 50:50 chance that you could get away with booking a AAA discount without a AAA membership. But I have no idea what your odds will be at the front desk since I've not checked in as a AAA member yet.
I suppose the real questions are for you - Do you know how you'll respond IF they ask to see your card at check in? Would it be embarrassing or would you take it in stride? Only you can know how you'd really feel about this.