If the website says you should get full credit if you cancel, then I don't understand why you would not just do that or why AA would refuse
I am missing something here.

UPDATE: The website says:"If you’re unable to reschedule your trip you can cancel it and apply any unused value toward the purchase of a new ticket within 1 year of the original ticket date. A difference in fare may apply."
What did they say when you told them the website said this? There is no mention it doesn't apply to BE fares.
Option 3.. Cancel, request refund from AA... Keep copy of the website advisory. Inititate credit card chargeback