Welcome to FT, DrSuess.
It's an AA ticket, so you'll have to deal with AA. It may be that the fare class you initially bought doesn't allow any further upgrade other than paying the full fare difference (which is what I would expect). I don't know why you think you would get it for less...
Your only other option is to ask at the airport; often there are special upgrade fares for less than the fare difference, but I wouldn't get my hopes up. WT+ is a small cabin and is regularly full.
I've alerted a moderator to move this to the BA forum for you.