It does sound like you got an agent that doesn't understand what to do
First, obvious question, does is there availability on the BA flight for a 1st class award booking? if not, then no point continuing further ; check for award availability on the BA flight

If there is availability, then the agent should be able to process an upgrade. You will get rebooked onto the BA flight number as opposed to the AA codeshare flight number

If you get an agent that is unable to assist, just hang up and try again - perhaps try a different country call centre