4wd and camping gear

For a three-week road-trip through both islands, I would just rent a medium or large sedan (or station wagon), rather than a 4WD - I have been advised there are very few roads in NZ where a rental car is prohibited from using. And for three weeks you could buy basic new or second-hand tents and other gear in Auckland. We are renting a sedan in December-January via a company called Shoestring in Christchurch - don't know if they have rentals starting from Auckland. They have a link on the YHA Hostel (NZ) website.

4wd not necessary at that time of the year, and the marvel of tarsealed roads has reached NZ. We even have a motorway or two.
When you get to Auckland if you are staying in a hostel, they all generally have noticeboards, & you'll likely be able to pick up some stuff 2nd hand from other travellers.
If not, there are plenty of good outdoor shops - a big one is on Hobson St, & usual suspects Bivouac & Kathmandu etc are in CBD.
Last time I was back in NZ I hired a car from Apex & it was fine.

and once you are in new zealand - trademe.co.nz

You can bring your camping gear from Belgium - it just has to be thoroughly cleaned as we are fussy about soil & grass seeds etc.