Just wanted to add that we did this about 30 years ago, last 'big before kids" vacation.
Rented in Christchurch and returned. 12 days around south island- headed west to the glaciers, then counterclockwise- down to milfrod, dunedin, then up to queenstown (as we drove into QT there was a bungee juming group on a nbridge. ground zero is it turned out. I think it was $5 per jump

)
No advice on rental places- I expect they are long gone. HOWEVER, a short story- I did a bunch of research (may even have been phone calls and faxes) and specifically wanted a lower mile camper. Was assured it was very new, less than 60K kms. We arrive, it is near 5pm and raining. Van has 340k. Not happy, told that is all they have. Off we go.
First night, water is coming into the van, we abandon ship and check into a motel.
Next day we are headed to franz joseph and I call the company. Complain. Tell them we are abandoning the vehicle and will file w AMEX. They have me call back in a few hours:. They are sorry, ask "where will you be tomorrow AM?" I tell them 'camp outside wanaka'. Hold. "Our driver will meet you there with a new camper"
and he did. The guy drove like 350kms overnight to get there in the morning.
What a memorable trip..... (also spend a few weeks in Fiji)
enjoy!