So it seems the best way out of my employment tangle is to become a contractor and do contract work for my US employer (see previous question if you are curious of the whole backstory).
I don't think I can become a contractor in NZ with a Work Exchange visa (though I would love to be wrong about this). I am not looking to become a permanent resident, so I don't think the Skilled Migrant visa is for me either.
What should I apply for?