That's definitely doable in 3 weeks (we just went from Wellington to Cape Reinga in 10 days so driving-wise it is fine, we only had the one child in the back though) but just wanted to echo what is being said about it being more a road trip. Motels have all the kit you need generally to make meals and self-cater and I think they probably have washing machines. You should have no problems turning up and booking accommodation in July (unless they are perhaps closed for the winter, I don't know). We have never booked anything in advance (in mid-January) and haven't had problems finding somewhere reasonably-priced.
As for going to Nelson, there can be some reasonable weather days in July but being on the Cook Strait in the middle of winter could be a life time experience you will never want to repeat. Perhaps looking at airfares may be an option. They don't call it the Roaring Forties for no reason.
Exciting plans though! At least if you come in July you see the weather at its worst.