Depending how you are travelling, one option to consider for getting up to Arthurs pass is the Trans Alpine train, if you start from Christchurch, you can hop off at Arthurs pass or Otira and you are almost as close as you would get with a car(disclaimer I haven't actually hunted around these parts so someone may be along shortly to correct me). After you have hunted around the area, you could get the next train to West Coast and pick up a rental car from the railway station in Greymouth.
Another option to consider for travelling is this website:
https://www.transfercar.co.nz/
It is all the rental cars that need to do an empty run to get back to their home depot. More than half our tourists drive south instead of north in rental cars, so there is always a pool of free trips going north, sometimes going the other way too. If you are flexible about day and type of car and happy to bridge the gaps with public transport it can work really well, I use it a lot myself and really rate it as an option.
It won't cover your whole trip, but it could save you some dollars if that is one of your priorities and you can make it work.
I have a funny feeling that rental vehicles aren't insured for the Rainbow Road, but driving through there is a couple of days well spent if you get the chance (it is 4WD only), there are several DOC huts along the road that are vehicle accessible and deer in there too and you could tack on the Molesworth as well after a soak at Hanmer if you start the Rainbow from St Arnaud.
The South Island is a great place to travel, I spent a few days in Snowdonia in Wales a few years back and it reminded me very much of our South Island High Country.
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