British Airways doesn't fly between Wilmington and Baton Rouge which is why you can't book the flight as a cash/revenue ticket on ba.com

The only thing you can do on ba.com is search for Avios redemption availability on AA and use your avios as payment. You can transfer your Amex MR points over to BA to complete the transaction. AA isn't a Amex MR partner which is why you have to transfer to BA (BA being a partner airline of AA).

There does currently appear to be one-way redemption availability on BA.com ILM-BTR on 21SEP (via CLT) but nothing returning on 25SEP.

You can always check out Delta SkyMiles program for redemption availability too as you can transfer your Amex MR points directly to DL That might be another string to your bow and give you another availability avenue.

Also, ave you tried booking the AA flights on Amex travel booking site? That way you should be given an option to pay using your Amex MR points, although you'll be paying far more in the way of MR points for the privilege of securing revenue space as opposed to redemption space for the flights you need.