There are actually two SkyClubs at SFO: first, the one we are all familiar with, which is behind and one level up from the Delta ticket counter, outside the security entrance to Gates 40-49; and second, the former NW World Club which is beyond security inside the International Terminal.

While you could take the "AirTrain" over to the International Terminal, I've found that is just about as easy to walk (about 10 minutes from Delta's domestic operation).

According to delta's website, if you have a BE ticket, you have access rights to the Air France "Salon" and the China Air Lines club, which appear to be near the Delta club in the international terminal. I checked and neither China Eastern nor China Southern serve SFO.

I think if I were you once I get off the plane at Delta's domestic gates, I'd head straight for the international terminal, get through security, and relax. You could do an airline lounge sweep and give us a full report