For remote camping experiences I would recommend DOC campsites I guess. Local authorities have different policies for freedom campers on public reserves. Most of them tolerate (some even welcome) self-contained campervans i.e with their own toilets. Facilities will be lacking but you might get them to yourself. Try checking out district council websites for the reserves where you can do this, and what the local regulations are. Here in Dunedin freedom camping is tolerated - it's not permitted, and council staff can wake you up and move you on, but if you are responsible / clean up litter / use your own toilet etc a blind eye will likely be turned in most cases. As always, it's the irresponsible few that spoil it for the majority.
TBH though camping is / campervans are really popular in NZ. On several trips we've pretty much had the roads to ourselves while driving somewhere, felt in the middle of nowehere, then when we get to the camping area they are really busy - especially during the holidays.