I think this will be very difficult because Barclays' contract with AA does not permit them to accept applications, but perhaps there is an exception for existing cardholders of which I am not aware?
I had gotten my first card in early December, and I tried to get my second card on my 91 day mark, when the application was still available. There were reports of people being able to grab a second, so I thought I might try as well.
So yes, Barclay's contract doesn't permit them to accept new applicants for the AA card, but this was before that clause came into effect.