Here's a related thread from a year ago: Involuntary changes on a OWE itinerary.

Are you ticketed? Having made payment is not the same thing as being ticked. If so, you should have no problems. I'll assume your trip has actually been ticketed.

Basically you will need to be re-routed to CHC through SYD, WLG or AKL. I'd personally use AKL as the flight options are better.

There should be no need to bargain, simply contact BA and arrange your travel MEL to CHC to suit your needs.