Any airline flying this route will have business class.
I'm not positive, but I think the big surcharges on BA are only for flights into LHR. So if you can fly AA into LHR, I think you can return on BA and the fees won't be too bad. You'd have to check though.
If you are insistent on 777, you will probably have to fly Air France. It looks like BA and AA fly 767's out of JFK, and United flies 757's out of Newark. You could exchange SPG points for Delta miles and book on Air France.