We did that in 2015, it was great. We rented a camper van, most are diesel which is good there. It was a Mercedes camper much like the ones here but kinda stripped of luxury items. No problem though. We had it for a full 3 weeks and it cost about $4500 dollars, so not cheap. I would suggest getting one that you don't have to make the bed down every night and then put it up in the morning to have a table. (from experience)
Lots of choices, as seems like soon as you leave Auckland every other vehicle you meet on the road is a campervan rental.
Here is a link to check out rentals.... https://campers.vroomvroomvroom.com