A friend of mine was looking into trying a stag hunt in New Zealand. Im sure its out there but I told him I would ask you guys. He said that he couldn't find what the cost or procedures are for taking your own gun and ammo. Or what if any restrictions on it. He wants to do a public land hunt so I'm sure theres more to it than if you went with an outfitter on a preserve or such.
Hunting in New Zealand is AMAZING! However, most of the land in NZ is privately owned/controlled so to hunt "public land" you will be quite limited. More on the South Island but there you will also encounter more weather changes. Bringing a rifle is not much different than any other rules here for firearms, declare, locked case, ammo and rifle separate etc. There are so many other things to understand like flight weight limits. When they say 50#'s..... they mean 50#'s, to the point that when I was returning from NZ, (overweight) one of the check-in Nazis pointed me to an area where people repacked and threw away clothing items to meet the weight limit. There were scale areas all around the airport. You can not bring any meat back into the US from NZ. Without a connection in NZ, I would think that it would take several trips down to get the feel for the land as well as finding places to hunt. I stayed with a local and hunted, fallow deer, turkeys, peacocks and wild goats with him as he had access to several farms and I paid an outfitter to hunt Red Stag. Best hunting that I have ever done to the point that I am hoping to go back again next year.
This is where you go for permits and public land hunting areas: https://www.doc.govt.nz/parks-and-recreation/things-to-do/hunting/ Dig you way through this site, It will tell you where you can and can't hunt. Its free and easy to just print yourself off a permit, year round. Fiordland hunting is epic for large wapiti (elk, gifted to NZ by Theodore Roosevelt), allot of that hunting has a season and is by ballot. And as Elk guide said the weather can change pretty quick down there. I been there a couple of times but not to hunt. More infor here : http://www.nzhuntinginfo.com/game/elk.html Cheers.