There are a few legally permitted places around Kulima Estates - basically you need a NUC permit. If the owner won’t provide that number to you, it’s a safe bet they don’t have one!
http://www.honoluludpp.org/Portals/0/pdfs/HotIssues/NUC2018.pdf - this list of ALL the legally authorised rentals on the whole island.
Have you visited Oahu before? Have you stayed in Waikiki? As a regular visitor it always surprises me that people want to stay at North Shore and Aulani. IMO they are very similar, so you are swapping like for like.
We always split our visits however we do Waikiki / Ko Olina - for our family it is then two different holidays. Ko Olina is being lazy and staying put whereas Waikiki has a mix of great beaches as well as attractions, dining and shopping.