I would just wait for a response. Every time you contact AA, it is effectively resetting the "clock" on your submission. Calling will not make this any faster or slower. I certainly would not expect to hear anything for a good 45-60 days from your last submission.

You could complain to DOT (onlune and cost-free), but that is frankly not going to help here in that AA will simply respond with the same documentation it provided to Amex.

You could also sue AA in small claims court. It is unclear to me where you are located, as your profile does not give a location, but one suspects the UK based on payment in £ and BAEC status. Thus, small claims is likely to be an unworkable solution.