Unfortunately I"m tied to LHR & BA. I"m flying direct.
See...this is gonna bite you in the posterior....I don"t know why you are tied to one airport and a single carrier and no stops but if that
IS the case then you"re just going to have to take what they give you.
By adding just a little flexibility it is
possible that you could save some money. You might want to look into BA codeshares as another option.
As for one-stopping, you have multiple options, Adam mentioned one (LAX) but don"t discount San Francisco (SFO). The flight time is about the same as to LAX and then the internal flight is only a few minutes longer.
If you have the time, the drive from LA is a fun one (but long)...budget at least 6 - 7 hours depending on traffic. An overnight stay somewhere in between can make it sort of a road trip.
Otherwise, you could do the one-stop flight anywhere (Chicago, New York, Dallas), you just have to add in extra travel time.
I"m not saying that the savings will be huge (compared to the extra travel time...probably 4-5 hours) but it"s an option.