Consider using your AA points to book a flight on BA.

Looking at October, BA has at least TWO business award seats for LAX-LHR available on 21 out of 31 days in the month. You will have to pay some fuel-surcharges to BA as well as a UK premium cabin tax (the business-class tax you'd have to pay on AA as well), but going BA is the way to go.

You will find BA's J experience far superior to AA's.

To further illustrate BA's availability, I looked for JFK-LHR in July, also in J. They have non-stop availability open, in business, every day of the month. July is typically a tough month to get TATL awards - but with BA, you pick the day...