Finally, AA have given me the points back without a fee. I went onto Facebook, and they reacted swiftly!
The sting in the tail is that their reply says:
You’ll need to request the taxes be refunded in person at the airport. Make sure you give the agents at the airport your record locator or the ticket numbers.
I have cancelled the flight, so will not be visiting the airport!
The flight was with BA, although booked through AA airmiles.
Anybody know what I do next?