I spent a few weeks in the North Island in 2023 doing various routes. I wouldn’t rule out going north of Auckland for a bit: the Heart of the Octopus (250-ish miles) takes in the Bay of Islands and the kauri forest on the west coast and is a pretty easygoing introduction:
https://ridewithgps.com/routes/36144316
I’d second the comment about NZ drivers being terrible – all the advice I had from experience riders was to watch out. The further you get from civilisation, the more redneck the attitudes. I find the driving standards much better in Australia.
Re the Queen Charlotte Track and other popular offroad routes, such as Old Ghost Rd, I think you will have needed to book a spot in a hut months ago. The official routes in national parks are very popular and tightly managed. But I don’t know for sure.
NZ Great Rides website has loads of excellent routes of varying degrees of difficulty. Gravel routes like the Molesworth Muster look doable on TJ’s new bike. And of course there are the Kennett Brothers books.