Coming back to this thread again. Let me ask one more clarifying question again:

Using UA miles, which airline offers the best C product for the longest leg of the SIN-/connection/-AKL trip? The most important is sleep quality.

Second question - what would be the best connection points? I am ready even to fly SIN-NRT-AKL to make sure that last leg fits the requirements above.

And do I understand that checking UA's site for NZ or NH availability is useless?

I've checked wiki page for routes in AKL. NZ fies to DPS as well, however it is listed as seasonal route. SIN-DPS-AKL, if NZ offers C in DPS-AKL route with flat beds would be the best option...