AA only markets the full-fare "W" fare bucket on all BA flights - this has been bemourned on this board for ages now...you would need to buy a BA-issued ticket for access to discount WT+ fares.
When nothing is offered online through BA's MMB site ('manage my booking' - you do have the BA record locator and used this to assign seats etc. yes?), then, unfortunately, apart for hoping for airport cash upgrades to be available for about $250, you're out of luck. Ask for these at the airport ticket counter when checking in, and when nothing is available right then ask for the upgrade request to be noted in your record and try again with the gate staff. As WT+ passengers' upgrade requests to business clear, seats may become available.