I am a Kiwi living in Seattle and visit family every year over Christmas / New Year so I could give you some pointers on DIY hunting in the lower South Island.
It is very easy to take a rifle over there and the hunting regulations are the same for resident and non residents. Simply stated there are no tags required and no seasons on any species with the exception of some areas during the rutting period - mid March to mid April.
Have a look at this web site, register and ask for suggestions.
NZ Hunting and Shooting Forums
All that trophy hunting stuff, shooting of "free range" managed animals on privately fenced properties that you see on TV over here is the exception and does not sit well with me.
Here is another link to the official regulations about hunting on Public Land. The information covers everything but you will be amazed at how few regulations there are.
The only mammal native to NZ is a bat. Every animal that you would be likely to be interested in hunting was introduced for that very reason - sport !
There are no predators or snakes in NZ so you are safe to sleep out anywhere - but keep an eye on the weather.
NZ Police site for Visitors Firearms Permits.
Visitor's firearms licence and import permits | New Zealand Police