Basically no way of using flexipay on anything but ex-NZ tickets last time I checked, cash fare probably the best option.

Have you already booked/flown an outbound? There's some pretty attractive LH fares ex-NZ that you could use to do this.

I'd also try and get on SIN-CHC if your dates line up to avoid AKL altogether, unless you need NZ stock for an RU or something which is probably $$$ more anyway.