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).