Originally Posted by
Too much travel
You're right, though it depends on the time of year. On the winter timetable, the later BA flight goes off closer to 8pm, and the CX flight goes off before midnight so one can lounge-hop.
If the OP is talking about the morning CX flight, then perhaps the JAL lounge might be an option.
Unless recently changed, BA would only allow access if OP has OW-SA/EM status. The SFO BA lounge is usually crowded anyway. I would suggest OP not to go to the airport too early.