I'll second the Qantas First lounge. Just keep in mind that the Qantas First lounge is in the Tom Bradley Terminal, aka TBIT, while the American Eagle flight leaves from the remote terminal. That means you need to give yourself an extra 45 minutes to walk from the QF lounge to the bus gate at terminal 4 and then take the bus to the remote terminal.

Also, when you check in, go to terminal 4 checkin (not terminal 5) if you are going to the Quantas First Lounge.

I always get it as a QF Club member at the Remote AC. The remote AC is a fairly typical Admirals Club you can get into. Better than sitting at the gate but not remotely in the league of the Qantas First or Flagship lounges.