This does route through PER for a stopover if booking open jaw AKL/CHC
Fare rules allow 1 free stopover in Europe and 1 in Area 3 Asia / Pacific. Have not managed to get it to stop anywhere else than PER and DOH.
Adding 200 GBP for CPH, I've priced out at 2400 GBP:
CPH - DOH
//
DOH - PER
//
PER - CHC
//
AKL - DOH - CPH