Thanks. Looking at AKL-CHC, and the JQ flights aren't well timed for our purposes.

TK's in there as a joke - I haven't flown TK in years and never had many miles to begin with. In fact, Star is the one alliance we have close to zero contact with - hence the need for an AMEX transfer partner. I managed to search on VS, but as you say doesn't seem to be any availability.