We hired a campervan last year.
I really would not worry about driving on unsealed roads, we drove on several and found them in a lot better condition than many of the sealed roads in the UK - you really don't need a 4x4 to access them - at least in good weather, wouldn't fancy trying to negotiate a snow drift though. Just take it slow and steady and you will be fine.

On the South Island, there were only two roads that we were told we were not allowed to take the campervan on, one was the one leading to Mount cook and the other a canyon road and apparently even the locals avoid those if they can.

Or you could get the best of both worlds and buy something like a bongo which you can get a 4 wheeled drive version but you can also sleep in it and the older models can be had very cheaply on trademe.