You can have a stopover and an open jaw. So you could fly to BNE (or CNS) on your award, then pay your own way (or use non-DL miles) to wherever you want to be in NZ and not resume your award until you begin your return, so you could do AKL-SYD(stopover)-LAX. Do actually look at how far apart Australia and NZ are. It's 3+ hours across the Tasman (at least in terms of block times for flights). If you've only got two weeks, I think picking Oz or NZ might be preferable to trying for a lot in both. A reasonable alternative would be ~3 days on the east coast of Australia on the way to or from NZ. I find NZ is amazing, but it does take time to get around. The roads are windy and not flat. There are about 12 million one lane bridges on major highways on the South Island alone. The NZ AA gives a good idea of driving times between cities, but they're generally without stopping to take pictures or walk up to see a spectacular view.
In terms of domestic flights, they can be found, but make sure you only book low. I got OMA-MSP-LAX in F to connect to LAX-BNE in J, and there were actually multiple seats on all those flights. Be willing to take the flights to LAX in Y if they're available.