Your best bet is arguably SQ (Singapore Air) via MR+UR out of SIN, combined with AA on CX (or even mixing AA in itself to HKG) to get you to SIN. SQ has decent availability in F in their own program (since as mentioned, CX does not do F to Australia), and MR and UR are both SQ transfer partners (and there's a discount for online booking).
You can't book this as one award on AA as you are intending to do- AA does not allow Asia Zone 2 (HKG) transit to Australia/NZ, as mentioned. 2 QF seats on one flight from the USA on an award is like trying to buy a winning lottery ticket (good luck). North America->Australia/NZ in premium classes is probably the hardest single award to find out there. QF elites will have access to those seats before you do (QF gets them at 355, AA gets them at 331), and will snap them up promptly.