What you are looking for is OTL's Tonga, currently ruled by an indigenous monarchy and with a vast majority of indigenous inhabitants.
IMO to achieve both of OP's goals in other islands, you need to have a situation where the Polynesians control the land. Had the Hawaiian monarchy not ceded control of much of the land to Europeans, said Europeans would not have imported the thousands of immigrants who irrevocably outbred the native Hawaiians after the latter's population cratered from disease. Even if Hawaii became a protectorate of the British, the British would not kick out the white landlords or prevent them from importing labor and Hawaii today would have a similar ethnic mix, albeit with some native Hawaiian nobility being much richer than OTL.