Is the Aston Executive Centre in downtown still open? It's outside of Waikiki, but probably not a first choice for leisure travellers. Hotel St. gets kinda seedy after dark.

I'm not sure if the Ala Moana Hotel qualifies as out of Waikiki (warning - I saw some grim reports on TripAdvisor) - I might argue that the Prince, W, and Diamond Head Beach Hotels are both far enough outside of the Waikiki to be options as well, but you'd be driving many of the same routes as if you were staying at the Royal Hawaiian in and out of Waikiki.

The Ihilani and Turtle Bay are both good non-Waikiki choices - I think you've been given your best options by Mary2e and slippahs.