NZ & Aus Routing Question
 

Which of these are possible to do:-

LHR-SIN-MEL-CHC-AKL-SYD-JFK (MEL=transit only, but means going back to AUS from NZ on this routing isn't allowed)

LHR-HKG-AKL-CHC-AKL-SYD-JFK (AKL first time would be transit only) - stays would be in CHC, AKL & SYD.

Looking to max out miles possible whilst not doing unnecessary flights due to medical issues.

Both are allowed on a OWE fare that does not originate in Australia (you aren't allowed multiple entries into the country of origin).

And your comments about transits are irrelevant - the only limitation on transit vs stopover is in the continent of origin (and in the case of transits through Europe to Africa and N Am to S Am). There is no limitation on flying across the Tasman Sea (Aus to NZ) multiple times unless the ticket originates in SW Pacific.

Sometimes the webfares for AKL-CHC are really cheap, I did it round trip for $150 NZ. ($75 US?) So you could buy AKL-CHC-AKL seperately and use the freed up segments for something else.