Check if any of the properties you're interested in are listed by agencies, contact the agency and tell them what you're looking for and explain the dog. We have a brand new 3bd house on the North end of Upper Hutt that allows our dog and we pay $400. The commute is slowly killing me though, especially with the trains being out all this week.
Petone could be an option. A little further out, pretty easy commute by train (or bike) and still plenty to do, but it does still seem pretty popular so potentially not a lot of savings in $$$.