We are about to do a NZ caravan trip over the summer so I hope it is achievable! Will let you know how it goes.

Would you be going to North Island or South Island? Or both? How long would you go for?
North Island has two ski fields on opposite sides of the mountain, open from mid June until around then end of October officially. Best bet for skiing would be August/September. If you go to the North Island one there is a friendly BBer who has kids snow pants and boots from size 6 down that you could borrow There are plenty of hire places but might save you a few dollars.

For skiing with kids, the most important thing is good gear. If they aren't waterproof enough then they have fun for about 10 minutes and then they are wet and cold and start to grizzle. There are toboggans for hire which is lots of fun.

Rotorua is a good place to go. There are the built up commercial thermal parks, but there are heaps of free access mud pool around the lake and parks too. Gondola and luge are fun, there is also one at Queenstown.