Cycling New Zealand - Leaving Invercargill

I think you’re going to need to amend your plans a bit. Between Mt Christina and Kinloch there is nothing remotely rideable. The Routeburn Track is the main route across there and it’s a pretty rugged hike. The Greenstone and Caples tracks also cut across, but I’m pretty sure that bikes aren’t permitted on any of them, even if they were rideable. There are no roads of any kind across the Southern Alps there. There is also little to no infrastructure other than the road itself between Te Anau and Milford Sound. Outside of the summer season (October to April) this road is often closed at short notice because of avalanche risk.

The road from Kinloch to Queenstown would take you through Glenorchy onto the Glenorchy-Queenstown Road which I would strongly advise against cycling. It’s narrow and twisty and has lots of distracted tourist drivers. I’ve never seen a cyclist on it, with good reason.

The bike friendly route alotronic suggested is a much better bet. It takes you towards the relatively new Around the Mountains cycle Trail, you can pick it up at either Mossburn or Lumsden. This is mostly off road, has a decent surface and should be great on a gravel bike.

https://www.aroundthemountains.co.nz/about-the-trail/

It finishes at Walter Peak Station where there is a renovated steam ferry you can catch directly to Queenstown. Once in Queenstown there are a stack of nice trails around the district to get around on.

I cycled down through NZ from Auckland to Dunedin a few years ago on a touring road bike. Did not have a single problem with drivers. Not saying that there is bad driving but in in my experience no worse than the UK

A couple of starters for you:

https://www.opencyclemap.org/?zoom=9&lat=-45.55065&lon=168.82089&layers=...

Shows a bike friendly route to Lake Te Anau - I beleive (though you need to check) that you can connecrt the end of this route to Queenstown by ferry.

Also look at the Tour Aoteraro website - this is an end-to-end gravel event by the masters of NZ terrain, the Kennet Brothers, mostly done on MTBs or chunky gravel bikes.

http://www.touraotearoa.nz/p/map_22.html

The downloadable maps etc will give you clues on good routes - bear in mind that off road in NZ can get very serious very quickly, if you are going off road take the biggest knarliest tyres you can fit.

Also the Kennet Borthers do a huge range of MTB books with various grades of route - mouth watering frankly.

https://www.kennett.co.nz/ 

Finally if it was me I would be trying to stay off main roads as much as possible (I am a kiwi) they can be narrow, fast and there are a significant proportion of the population with big d*ck problems.

Having said that it is a fantastic place to ride and visit and, with a bit of common sense and a taste for adventure, you are going to have a blast.