We used to have a member who actually lived in New Zealand and offered to help any members who wanted to go over to NZ and hunt. As I recall, most game species most of us would be interested in were regarded as non-native to NZ and therefore could be hunted year round without bag limits. Unfortunately I don't recall that member's name and no longer have a few PMs from him (it was over 10 years ago). From what you've described above, I think he would have been an ideal person to talk to.
You can check out topics for New Zealand hunting at our regional forum.
https://www.huntingnet.com/forum/other-areas-29/
There are some members who have hunted NZ and they should have a lot of good info to share. BuffyBR hunted on the South Island I believe. Hunt genious also hunted down that way. Another member you might try to locate is AorakiGuidedHunts (Simon) who is a guide in NZ. His website back then was
http://aorakiguidedhunts.webs.com/.
Hopefully some of this is helpful. Your proposed move sounds like a fun adventure. If you do make the move to NZ, be sure to stay in touch and keep us posted about hunting down there. For many of us, hunting in New Zealand would be a bucket list type of hunt.