Thanks for the advice, all. I did try going into Paris or Barcelona, but couldn't find award availability (4 in coach, or even business, just not there.)

However, after my 5th or 6th call and different agent over 3 days, I did end up with this, rather creative routing:

Same outbound, AA PIT-ORD-LHR, 10 hours to kill in London, then LGW-TLS on BA.

On the return, on BA metal: TLS-LGW-BDA, overnight in Bermuda, then AA BDA-JFK-PIT.

There are worse places to overnight - I am going to try and see if we can stay two nights, if I can ever get an agent on the phone again!