The problem is that you can get WT+ fares from AA reservations over the phone if you ask for them and get a competent agent. These would not be eligible for an upgrade with a SWU or miles+copay, but they are eligible for an AAdvantage Participating Airlines upgrade.

However as the OP's experience indicates and BrewerSEA's research verifies the fares AA offers for WT+ on BA are impractical and not worth booking.

I would recommend booking either the CW or WT+ tickets through BA.