America's not going to tolerate a British protectorate; a possible alternative might be a Condominium along the lines of the New Hebrides (Vanuatu) or the Tripartite German/British/American rule over Samoa.
Actually, a German-American clash over Samoa could be a way to handwave America's toleration of British influence in Hawaii, as a necessary concession.
That system, if it's anything like Vanuatu or Samoa will be extremely inefficient, expensive and will have no buy in from either government. But just as Britain and France somehow managed to keep the co-rule of Vanuatu going until decolonization despite various attempts by Britain to disentangle itself, I can picture Hawaii staggering on year after year, with every new American 'advisor' to their Royal Hawaiian Majesty convinced that any day now Washington will be able to annex the place.
Crucially, that's a system which is going to discourage settler colonialism; Vanuatu saw serious concerns in Australasia every time the demographic balance tipped ever so slightly to the French. A Hawaiian Condominium will firstly encourage expansionists in New South Wales, Queensland, Victoria and New Zealand who wanted a more muscular British policy in Polynesia- in our timeline, after the Hawaiian monarchy fell the New Zealand premier threatened President McKinley with dire consequences if America moved in.* For America's part, there will be less interest in emigration to the islands given that there will be much less easy pickings and less support from the government.
Throughout all this, I'd expect the monarchy to retain a nominal role, aka to the Kings of Tonga under British influence.
How this Condominium would deal with Japanese or other imported non-white labourers is hard to say. But I can envisage this leading to a majority-Hawaiian state at the time of decolonisation, albeit with an American 'lease' on Pearl Harbor.
*I also have a scribbled reference in my notes to him suggesting that Britain try to enlist the aid of the Japanese to keep America out of the islands, almost a decade before the A-J alliance!