Camping near crags is relatively common.
Do you know approximately what route you'd be doing?
Wharepapa South would be an obvious stop. It's the biggest collection of sport climbing on the North Island. It would be relatively easy to keep an eye on your stuff while climbing. Bryce's is close and rents everything you'd need. He also has all the beta and a few rooms to rent. You could also camp at the school around the corner.
https://www.rockclimb.co.nz/hire-gear
https://www.wharepapa.school.nz/camping
I have seen people bicycle to Kawakawa Bay relatively often. The approach is along a mountain bike trail and is long enough to warrant camping for a couple days. I would not do it on a road bike, but you could rent a different bike in Kinloch if needed.
The Port Hills around Christchurch have a bunch of crags where cycling would be a day trip from town. Not much camping at the crags though.
I would guess that your route passes near Wanaka. I'm sure you'd have plenty of options there too. And maybe Queenstown.