Ah ok. Did wonder as if you do it the other way around and book with BA having an AA flight BA MMB does list the AA PNR when you click on the AA flight.

Unfortunately third party tools like checkmytrip won't be of any use here since although Sabre bookings (AA's system) are generally accessible, AA doesn't allow third party access to their bookings.

Bit surprised at the issues you have had with BA and AA call centre. All you need from them is the BA PNR which either call centre should have been able to give you. Once you have that you should be able to access it on BA MMB and select the seat.