I've just made some further enquiries about this, and this is the answer.
So far as I am aware only the dependent children of PhD students or an exchange programme or an Aid programme can be domestic students. I know it used to be possible for degree and post-grad students too. However, there is nothing to stop a child getting a place either at a state primary or secondary school if they have places available or at a private primary or secondary school. Fees at such schools are nothing like as high as the equivalent schools in the UK, so this might be a possibility.
If any more information comes to light, I will pass it on.
Have you looked into whether the course your wife is going to do would entitle you to a partner-sponsored work visa?
http://www.immigration.govt.nz/opsmanual/46524.htm