You need to set up an account with AA so that you can use their online tool (this is a bit like registering for BAEC).
Then you need to log into that tool and access the AA MMB equivalent.
This allows you to search for your AA (BA Codeshare flights) using your surname and flight dates and numbers. You need to use the AA flight numbers.
AA MMB should then allow you to access your booking and you can change your seats using that tool.
Although you log in with an AA account it does not seem to try to apply the AA mileage account details to your booking, which clearly shows the BAEC number.
Without an AA login you need to deal with BA, but you can obtain an AA login practically immediately online.