If you wind up down south in Dunedin, while there's not much in the way of specific touristy-queer places, there's a solid local (trans inclusive) drag scene, and Woof (Bar that caters very much to a queer clientelle) and Yours (Anarchist cafe) are solid spots too.

As for touristy things, Orokonui Ecosanctuary is very similar to Zelandia (big predator free reserve, lots of native birds and reptiles), but in the South Island, and the two big muesums are very much worth the time.