I have never hunted that area with a bow but I am a New Zealander who grew up in Rotorua. That is probably the large town that you may leave from to travel to your hunting destination.
I spent hours in those forests as a kid so know that area very well. Murapara is the small town or village that services that area (mainly logging). Lake Waikaremoana is one of the most awesome prestine lakes that one could lay ones eyes on.
New Zealand is great - Kiwi people awesome and you should have a ball. You did not say what you are after but I would think pig and deer mainly.
Urawera National park is a huge area, thick forest and lots of mountains to climb.
I find it hard to imagine how one could spot and stalk in those areas with the limited range of a bow - it would be like finding a needle in a haystack to locate game in such thick forest. However I have lived in Australia now for 20 years so may be showing my ignorance to what is happening there now as far as outfitters and ways of locating game (baits etc).
I have travelled all over the world and suffice to say (though I may be biased), New Zelanders are some of the greatest, friendliest people on the planet. So whatever happens I think you will be given a fair deal.
Keep us posted - Sutto