I suspect your COL calculations are off because there are a lot of things you do not know that you need to factor in (I am not saying this to insult you). I lived near Wellington briefly and despite it not being my first international move there were a lot of costs that took me by surprise. NZ is so small and faraway that many companies do not ship their products there so the selection of things such as furniture, clothes, and electronics will seem much more limited than you are used to and the prices will seem high. I suspect (but don’t know) that might be even more of an issue on the South Island where the population density is even lower. In Wellington and its surroundings I noticed that domestic produce that traditionally grows in milder Mediterranean climates such as eggplants/aubergines and bell peppers/capsicums were more expensive than anywhere else I have ever lived. Colder weather produce was less expensive but if milder weather produce is a big part of your diet you may be radically changing how you eat to keep costs down. This can be a painful adjustment to make with young children especially because it is likely there are a lot of branded comfort foods/snacks they will already be missing. Another thing to keep in mind is that NZ salaries are lower, so without equity from a previous home sale in the US you may find the salary mortgage multiplier daunting.

Do you have family and friends you care about that you will be leaving behind? I have never felt as isolated from the people I care about as when I lived there and I have lived in seven different countries. It is a great move for some but it is not as straightforward as it seems.

I usually try to get a more micro cost of living that applies to me by looking up which businesses in my target destination provide the goods and services I want and see if they have a website that shows pricing. I do this for things like phone and internet service, groceries, cars, furniture, etc.