I stayed in Tingaki last year and spent a couple of days driving around the island.
I think there is a mtb guiding company on the island but they were quite expensive if I remember rightly. Would think they’re pretty essential though as the map I bought had nothing much in terms of official trails or anything. Not sure on the access laws either so a guide is probably the best bet.
In the end, the heat got to me and I just decided to be a lazy slob instead for two weeks!
If I were to go back, i’d either book a guided mtb trip or probably hire a road bike though as the roads are pretty quiet away from the main A roads and I think it would be more rewarding/less frustrating than trying to find my own routes.