I did this very connection in late January. The LHR flight was about 15 minutes early and I stationed myself near the exit door of the bus, which meant I got to Immigration faster than the other people on the bus. I found a line of about 8 people being served by two officers, so did not take long and my bag must have been one of the first off the belt (not relevant to the OP who is not checking anything), so it was waiting for me as I walked up to the belt.

I then re-checked (no line) and walked to T7 (LA sun after London snow was very welcome), went through security (no line) and arrived at the gate as boarding was about to start. I was not fussed about this as I knew the next flight was fairly empty so could easily have switched.

If I were the OP, I would reserve the later flight and then do two things:
1) ask at LHR to switch to the SFO flight - they will do this if there is space available, and we are returning to 2x daily.
2) if that doesn't work, when you arrive at LAX ask to standby for the earlier SFO flight.
Alternatively just enjoy the RCC - or the sun - at LAX