Perhaps, a situation not so extreme as the mentioned, USS Maine, but perhaps the situation, that the above mentioned Coup attempt or similar kind of incident, would spiral so quickly that somehow both Navies vessels, may have part of their crew on land and get involved, when/if a shooting would happen?
Or, even if, perhaps, less likely, could be happening, that the Australian Captain might be forced to take active measures to protect his ship and crew from either a mob/armed gang attempting to get/harm to the Hawaiian Royals under his watch (sames that are his government guests).
Also, I would guess, that, any possible involvement of the American ship, might be depending, on the American captain's reaction. When/if being asked to support and/or if he would witness that some of his countrymen are in danger/being shot by the Australasian... Now, in the broader context, I think that would be interesting to know, if TTL US and Japan, were able to overcome their foreign policy differences, about China and, if they're, IMO, likely sharing of interests in the Pacific (competition/opposition to Australasian ones), would turn, at least, in a de facto alliance between them...