You could try looking for a J award to SIN , KUL or even BKK, MNL, CGK or DPS, then see what budget/LCC flights you can get to Aus/NZ from there. (Air Asia, Tiger etc)

Then use stray points on OW or Star to get some flights around Aus/NZ.

We have done that successfully a few times. There's a lot better J award availability to the big Asian hubs