A bit of a longer route, but I took MU to AKL via PVG. Return from either MEL or SYD is possible on MU/DL.
For AKL-CHC just use NZ or drive/ferry as part of your trip. CHC>>MEL possible on VA as others have said or again may be easier/cheaper on OAL (we were going CHC-SYD and ended up taking EK).
If you can get the big TPACs on SkyTeam, the flights within NZ and OZ aren't too bad. NZ to OZ is a bit longer but still would rather get a direct flight on another alliance than spend time connecting around just to stay with DL partners.