This article from an NZ LGBTQ website does not answer your question, but it does talk about two LGBTQ friendly resorts (the resort they mention in Aitutaki has a tripadvisor review which also lists it as being Gay friendly in the title) as well as one (expensive) restaurant owned by an LGBTQ couple.

https://gayexpress.co.nz/2023/07/the...-our-doorstep/
https://www.tripadvisor.com/ShowUser...k_Islands.html