Certianly you will be able to get around, and certianly in the South Island it will be cold. I don't know how they go for heating those things, but imagine spending winter in a caravan. I think they have mains powered heaters but that requires you to be in a camp ground every night, which might remove the benefit of having a camper.

What about renting a car / swagon, and just staying in the backpackers? They will be pretty empty and much warmer.

The worst thing will be the internal dialogue that will run like a broken record "Gosh this would be fantastic in the summer!"