Quite ASB if you ask me. The Americans barely won on the land, engaging the British in the Caribbean is out of the question imo.
But the US Revolutionaries DID successfully launch a naval invasion, and win the Battle of Nassau against the British, in early March 1776. IOTL, they merely occupied the Bahamian capital for a fortnight, and took away all the remaining gunpowder and munitions which had been confiscated from the Province of Virginia and taken there by Lord Dunmore after the conflict broke out, before withdrawing peacefully and returning to Connecticut in early April. But what if they'd garrisoned those captured naval forts, and maintained an occupation force on New Providence instead?
In the immediate aftermath of Esek Hopkin's successful raid/invasion IOTL, the Continental Congress and individual state governors through their legislatures agreed to allow privately owned ships to help in the battle against Britain by issuing letters of marque. These privateer ships were allowed to claim any items found on the British ships they conquered as their own, and were therefore able to pay their seamen and officers nearly twice the amount that the Continental Navy could pay their crews, since the items captured by Continental ships went for the good of the colonies. Which, ironically, severely depleted the manpower of the fledgling American navy. However, this also served as sufficient incentive for thousands of these privateers, overtaking the number of British ships, which immensely helped the American revolutionary war effort at sea.
And of course, Nassau had always been primarily a privateer base. Sixty years previously, it had been declared as a Pirate Republic under Blackbeard's rule. And it was thanks to the wars in the Thirteen Colonies, such as the French and Indian War, and the influx of funding from its primary industry of privateering, that Nassau had experienced its greatest economic boom, with the proceeds enabling a new fort, street lights and over 2300 sumptuous houses to be built, as well as for the mosquito-breeding swamps to be filled in, greatly expanding the city. With the Americans electing to secure their hold over Nassau and the Bahamas ITTL, and issuing their letters of marque in the immediate aftermath (just as they did IOTL), it isn't hard to imagine that practically all of those privateers who operated out of Nassau and were employed by the British IOTL would readily switch their allegiances at the drop of a hat, and direct their efforts to raiding British ships on behalf of the American Revolutionaries instead ITTL. There'd have been higher risks, but also far greater profits to be made.
And as a result, the Bahamas could well end up being a core part of the United States from its inception ITTL; perhaps even a State in its own right. And if they had, there could be other butterflies as well. For instance, after American independence, the British resettled some 7,300 Loyalists with their slaves in the Bahamas, and granted land to the planters to help compensate for losses on the continent. These Loyalists, who included Deveaux, established plantations on several islands and became a political force in the capital; and from this point on, African slaves and their descendants constituted the majority of the population of the Bahamas. But without the Bahamas, where else do those Loyalists go- or stay? Would the British still be prepared to cede both Floridas to the Spanish as part of the Treaty of Paris, or might it refuse to concede one or the other of them? Or would those American Loyalists be resettled to re-establish their slave plantations elsewhere in the Caribbean instead? And if so, where?