Not to NZ but I used James Cargo to/from USA in 2014. Very good service at Manchester - New York - LA - and Manchester again. They also have a depot at Heathrow. I have heard good things about Motorcycle express which I think was set up by a couple of disgruntled James Cargo employees.
Also it might be worth a direct approach to whatever NZ air is called or indeed anyone that does a direct flight. (I know there’s at least one stopover).
I would be interested to find out how much it costs on the end Sean. NZ could be a destination for us at some point.