1. The Bermudians had a cross-section of colonists in rebel and loyalist sympathies akin to other colonies, so having rebels get into power somehow could already seal this. Barring
that, they were swarmed by American privateers once the Revolution began, and in 1779 with only ten minutes to spare the HMS
Delaware brought 100 loyalist troops and supplies to the starving island, missing four American privateers with many more guns. Have those ten minutes be lost, the
Delaware sunk or captured, privateers taking over the island (either in return for promised supplies, or just pissing off loyalist colonists into realizing the British cannot help them) and you can have Lord North write the islands off as another lost colony or conquered territory by America or whatnot.
2. Bahamas could have the classic marine raid of '76 be longer and whip up more people into rebel sympathy. Barring that, not many people realize that the Spanish capture of the island in 1782 also had American sailors involved-it was just the Spanish who actually landed troops and could claim the island.
Perhaps America can claim the Bahamians as 'fellow Americans' citing the English heritage and their part in the '76 and '82 raids. Assuming England doesn't have something to offer-Spain already captured Florida as well-we could have them backing American claims to the islands. After all, English and American ports will soon be open to each other again anyways-does it matter if the English-speaking Bahamians simply live under a different flag if they're still bringing in the bucks?
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As for the navy, remember that the ACTUAL American navy was not existant till 1798, but privateers very much active. Even if the island groups we speak of get captured in 1812's war (bloody likely) the resulting British occupations will get mercilessly harrassed by pirates. I see them being returned as part of the 1812 status-quo.
I also say these have a much better chance of happening for some reason if you can get Prince Edward Island and Nova Scotia into the fold...Lord North just gives up more territory since it's obvious, so to speak, so many English-speakers in North America seem to hate the motherland now.

(yes, I'm aware of the loyalists, but more American lands won means more likely just accepting their fate as-is)
(I'll also say regarding the HMS
Delaware: why did my homestate's namesake kill off more wankish possibilities for us? Yar!

)