Success story here. I had read that business class to NZ or Australia is the hardest to get with miles of any kind, so I studied up, like you are doing.

So when we went to NZ a few years ago, we chose to just use miles for the US-Australia leg, and buy tix to NZ from there (lots of options between Emirates, Qantas, and others). That kept is simple. We booked one way at a time as suggested above (this is important) and for the outbound I searched for flights from LAX to any of the 3 main airports in Oz (Sydney, Melbourne, or Brisbane).

We had enough AS miles for two tix in business class one way, and British Airways miles (Avios) for the other. (The advantage of the BA Avios is they get access to QF flights at T-355, a month sooner than they are released to other partners like Alaska.)

I started checking for the outbound flight on the Alaska award flight calendar every day beginning T-330. And one morning when I looked, I saw 2 seats in First Class on Qantas, LAX to Sydney, sitting there. So I booked those. 😊

Then I started looking for return flights from Melbourne using our Avios the day they were released, and got Business Class back to LAX.

After that, I booked our flights from SYD to Christchurch and from Queenstown to Melbourne, one on Emirates and one on Qantas. They were not expensive. One thing to know is that if you are flying into New Zealand on a one-way ticket, you may be required to show tickets for an outbound flight at check-in (Emirates required this and I was ready with printed copies of our Queenstown to Melbourne flight).