Hello,
I live in Gore, about half an hour from Invercargill. We are more likely to get cold weather down here than anywhere else, and the frost's can play a havoc on the road in winter, but honestly it doesnt get that cold. It's still warmer than home most days, and this year we had on average warmer summer days than anyone else in the south island. This year it has gotten colder quicker, wasnt really much of an autum, we have already had two snow's, but they wernt bad.
If i had to pick a place it would be Christchurch, not nearly as big as Auckland, and it has a great hospital, nice people, great shopping and everything you could need. Invercargill is alright, but it's a very small place, especially coming from over there, and there is only so much you can do there, also your schooling options are limited. But thats just my opinion, i hope everything goes well for your family.
Katie