I dont live in NZ or Australia, but have travelled there 4 and 3 times respectively in the past few years.
We did driving trips in NZ on each visit - both islands, North Island only and South Island only.
For 7 days driving trip in South Island, it is way too short - especially if you want to do a circle... We did a clockwise circle in 2007 in 10 days, and felt it was just barely enough.
However, if we were to re-do it, there is NO NEED to do a circle - the South of South Island really is NOT scenic at all if you only drive thru it - the beautiful spots and areas all require some hikings (miles long hikes) and you would not have enough time whatsoever in your allocated time.
The most beautiful parts of South Island are: Queenstown and its vicinity, and the secondary road going North from Queenstown, The Southern Alps, The Arthur Pass of Mount Cook, The Abel Tasmin National Park at the Northwest tip of South Island, The secondary road from Nelson to Piton, high above the mountain, offering breathtaking views of the passageways and inlets of the South Island... The Kaikoura Coast on the East side...
I were you, I would fly to Queenstown, and then do a ONE-WAY drive from Queenstown back to Christchurch, clockwise. You may have to make a choice whether you want to do Arthur Pass of Mount Cook, or opt for the Abel Tasman National Park, Nelson to Picton and then Kaikoura Coast... You could do this in 7 days, barely.
3 days in North Island - you would also have to make a choice - whether you want to go North or South...
Most Rental car companies DONT allow crossing from North Island to South Island. Even if it is allowed, the ferry fare for car crossing can be quite expensive.