Equally curious - the South Island has insanely cool country that reminds me a lot of Montana and Wyoming. Depending on the time of year you're there you should absolutely pack a flyrod
Ive been looking into it as well, seems like the biggest hiccup is getting the firearm permit and a sponsor in the country. Heard police stations are really good contacts with that.
I've lived there and hunted both quite a bit DIY, would be happy to try and answer questions. I also have some good buddies who would be great local guides if you were interested in that route. PM works.
A friend of mine is there right now. He has taken 3 tahr, 2 stags, 3 goats, a wildboar and some other stuff. He is having a ball. The keas in his camp are fat as butterballs.