Based PER rather than MEL - I chose to ticket PER-NRT(AA)USA(AA)NRT-SIN-PER to take advantage of the DBLEQ promotion. But I'm also taking 6 days to do it. The open jaw was the only way I could get a decent price on the N class fares. Would still work if the cheap flights can be scheduled around MEL(SYD)-NRT-USA.

Other than that ???
I can only suggest something on the order of AKL-LAX-DFW-ORD-MIA-Europe and return using as much AA metal as possible. About 18k Qmiles to FRA not counting MEL-AKL. Alternately BA in T class SIN-LHR-Canada and return will net about 30K Qpoints at a fairly reasonable price. But no status bonus on the redeemable miles.... And BA T class SYD-LHR and AA on to DFW would get more than 20K Qmiles or Qpoints each way with the promo (subject to a V fare class or higher on AA)

Be creative when you wander

Fred