Kiwibank wouldn't match my existing limit with ANZ--apparently I've hit their internal limit for people who don't do retail banking with Kiwibank. Slightly frustrating, but I digress.
Since I did a balance transfer with the new card (at 2.99%, a little arbitrage is in order), I'm going to avoid charging anything to this card for 6 months. But I see no reason for keeping the ANZ beyond the next 6 months since it is clear to me that the new Kiwibank cards are the premier product for Air New Zealand.