OTTM, you are already at one of the furthest airports (in the US) from LHR. You cannot better your "resting time" from any other US airport, not really. You're just increasing the hassle by adding a stopover . . . increasing your total trip time, but effectively reducing your "rest" (sleep) time because you've time-shifted it to a later time on a shorter flight. (Or, in the case of an SFO or LAX connection, you've added hours of domestic flight stress and won't rest well unless you actually stay over at your stopover before getting on your int'l flight.)

I'd shift my travel plans by one day forward or backward to get "availability" on BA0048 (and thus a full night's sleep on board -- in ONE flight) before I'd exhaust myself trying to position myself in the US to get a good night's sleep on a BA connecting flight.

That said, I did once consider a YVR connection because the connection would have been fairly smooth and not too awful. The return was terrible, though, and open jaw was overpriced, so the solution (for cost) was to fly out one day earlier.

But, if you're simply trying to meet up with an award flight . . . We feel your pain. -- I'd avoid JFK (or anything within 100 miles).