As others have said best chance would be BA LAX-LHR non stop but they will stick you with the fuel surcharge on your "free" award. But then that is why there is availability. Otherwise DFW flights tend to have higher inventory available to confirm. In my experience LAX -LHR and JFK - LHR have least pre book inventory but always got it at airport on JFK-LHR. AA137/136 LAX-LHR is probably the worst for upgrade availability. Remember the domestic leg will be domestic so even first class is pretty nasty (if you have just come off an 8 or 9 hour flight.) If it was me if I could not get 137/136 (ask them to send request to revenue) then I would look for BA non stop (there are 3 744's a day at present). Seats are parralel flat but not good if you are tall as they are shorter than AA with hard divisions at end of bed so you are cramped in (OK if under 6'0" or 6'2")