I've only been to NZ once for a month and didn't have enough time. With just a week, I'd concentrate on one area. If you are flying from the US, that will eat up a day or two both directions. Personally, a week just wouldn't be worth the long flight. If you aren't in an aircraft or airport, you'd be fighting jet lag.
With just a week, I'd concentrate on one area. Late May is winter; equivalent to late November in the US. The climate ranges widely from north to south with north being much warmer so I would focus on the north part of the North Island for your timeframe. There is so much variety, that you can quickly get overwhelmed with all your options. I'd pick one island and focus on that and then narrow down from there. Auckland will involve less flight times, and the weather is better so I'd go with the North Island. I was there in March (fall) and it was way too humid and hot for me in Auckland so doing that area so May would be much better. Save the South Island for a time when you can go in their summer/fall.
I did a quick check in Interval and found nothing for exchange in NZ for your timeframe, but there is a Getaway that might work. Check it out. Finding a 2-3 BR this late in the exchange options will be tough. However, there are lots of self-catering places; i.e. motels or apartments with kitchens. This is what I used for my trip and I used the NZ version of AAA as a resource. It would also give you more flexibility.
Sue
