The problem with such a late PoD is that the Kalakaua dynasty were somewhere on the spectrum between pro-American modernisers and the puppets of plantation owners. David Kalakaua's first major piece of work was the Reciprocity Treaty of 1875, which deregulated the sugar trade between the USA and Hawaii, allowing white adventurer-capitalists to graduate from being advisers and missionaries to being full-on monopolists of Hawaii's wealth with what amounted to their own private militia, the Honolulu Rifles.
These white incomers proceeded to impose the Bayonet Constitution of 1887. Under this Constitution, most of the native Hawaiians were denied suffrage (which they had exercised previously without wealth limitations since 1874) while resident aliens were given the vote. From 1887 onwards, all attempts by native Hawaiians and the Royal Family to check the progress of these plutocrats failed.
So the most plausible scenarios for an independent Hawaii would be pre-1874. The success of the various rebellions of Robert William Wilcox could plausibly reverse this process, but ultimately he was a quixotic adventurer only out for himself. Equally, the proposed marriage between Princess Ka'iulani and a Japanese prince weren't taken seriously by the Japanese and, as you will be aware, the Japanese weren't that interested in power projection at that time.
So here are some alternatives to the Kalakaua dynasty:
1) Lunalilo survives and has children: he wasn't married IOTL and he was never altogether healthy, so surviving the decade might not be all that realistic for him. Still, it isn't ASB. He was genuinely democratic, but he was also in favour of the Reciprocity Treaty until his advisers talked him out of it.
2) Queen Emma of Hawaii wins the 1874 royal election. She was pro-British and was the darling of the Native Hawaiians, but the Legislature was less keen on her and voted for Kalakaua. It was rumoured that Lunalilo favoured a popular election to find his successor, and if this had been carried out, Emma may well have been chosen. However, she had resolved never to remarry and died in 1885, so that might just delay Kalakaua's succession. The only close-ish relatives of Kamehameha I at this point who hadn't ruled themselves out were Theresa La'anui (wife of the aforementioned Robert William Wilcox), Mary Pa'a'aina (Emma's sister, she didn't have much of a presence, had no surviving children and was married to an American) and Albert Kunuiakea (bastard of Kamehameha III; he seems to have been popular and he was deeply mourned upon his childless death in 1903).
3) Prince Albert Kamehameha survives. he was the son of Kamehameha IV and Queen Emma, but died young in 1862. This might be a bit too early that you're willing to go.