The issue here is probably the stopover pushing you into a more expensive fare (though not necessarily a different availability bucket).
(Edit: it does book into R - not sure if the dates or routes would go into the free upgrade to First promotion).
Starting in LHR would be cheaper on two tickets.
Option 1 is to do it on two tickets like this:
You can then do NYC-SFO on a separate ticket - it's £719 one way on a transcon business fare, or £196 in main cabin, or £540 or so in first with a change in DFW (mainly).
Option 2 is to do it on one - but the start points are different.
Second edit:
ESFLYER95's routing is better - in terms of quality and service as well as flexibility. By booking an R-class ticket, you get (a little) more flexibility at essentially no extra cost over the two-ticket solution.