The difficult part is if you are looking for business availability, since it is obviously very difficult to get Qantas business awards LAX-Australia, is that it is REALLY hard to then also get QF J JFK-LAX for the same date. I don't believe AA's online award search will show an option for J on QF JFK-LAX and then Y on the TPAC leg, so this makes it even more difficult. It might be possible to look it up on ExpertFlyer but I no longer have a subscription there so I cannot verify myself.
FWIW IMO, more general advice, now that QF is 789 JFK-LAX, for those flying from NYC:
If flying Y LAX to Australia, fly AA A321T, 789 Y seat definitely tighter than A321T seat.
If flying J LAX to Australia, fly QF 789, the J seat is definitely better than the A321T J seat.
If flying F LAX to Australia, something of a tossup. My priority would be to make sure that I had at least a 90 minute connection at LAX to have dinner in the Qantas First Lounge at LAX.
The QF flight JFK-LAX used to have routine delays with the 744 when it was operating there; it seems the 789 is much more reliable from a dispatch perspective, so that consideration is no longer valid (I would have strongly suggested avoiding the 744 operated flight for that reason).