The problem is EK. For any other airline, you have tons of time. EK specifies in their C&C that economy pax must check-in 3 hours prior to departure time (and 1 hour prior for J and F). Technically they can unload you and deny boarding if you don't checkin 3 hours before; in practice this is enforced only if they are oversold (saving EK the expense of IDB compensation). Does NZ interline with EK on separate tickets? That would solve your problem. JQ does not, so you'd have to arrive earlier to meet the EK requirement.

As for CHC as an airport, domestic to int'l transfer in 45 min works fine (it is a small and efficient airport); I've done that several times (but on single ticket, so no risk in the connection).