Doable? yes. Violation of the AA
Conditions of Contract you agree to upon purchasing? also yes.
Without putting your AA number in the DFW-BHM-DCA itinerary, you certainly have less chance of a match being made. They may well use other identifying information like credit card used for booking, phone number, etc.
Worse case is they cancel your DFW-BHM segment. Worst case is they close your AA account, forfeiting all miles. Absolute worst case is they come after you for the lost revenue (99.9% unlikely to happen).
It should also be pointed out, in this decade, birthdates are also in reservations.
Another option for the OP would be to cancel the existing ticket and booking a new one. I assume the client will pay the cost for him to fly from home to where the client needs him