3 hours before your flight - https://www.qantas.com/us/en/qantas-...ort-guide.html
(if your flight is not the first of the evening you may find the desks open and the flight open)

Will you be checking luggage? As I assume that southwest will not check you bags through and you need to pick up bags and wander over that will probably take up the time.

If you are hand luggage only, can you walk on the secured side to Tom Bradley - I know you can from some terminals. As you could walk on the secure side, and then hang out in a lounge or AA Flagship lounge until checkin opens and wander down and get a BP. Unless one of the lounges or gate will offer to do that?

Online checkin should work unless your itin is complex or your visa complex.

KF