American privateers had a good shot at taking over both the Bahamas and Bermuda at one point. Keeping either colony would be huge in the development of America.
This is as far as I'd go with that as well, of which only the former can be counted as part of the West Indies. There were sentiments shared with the Patriots throughout most of the British North American colonies in some form and quantity, big or small. Sure, many of those weren't faced with enough reason to rebel, others with their minuscule populations had issue to fear slave revolts should they lose British protection, but the larger issue is still the Royal Navy. Their waiting bite would've made attempts to join the Revolution in places like Jamaica, far and removed from the mainland and the main concerns of the majority of patriots, appear seemingly suicidal to those both at home and abroad.
Perhaps though, to honor the challenge, I'll toss some ideas about even if they seem a reach to me. There was great opportunity in the French-Spanish combined fleets to do devastating work upon their imperial brother's (The Armada of 1779). If I am remembering correctly, there was even a massive gathered force between the two nations, looking to find and disable the main body of the Royal Navy. From what I understood they didn't really manage to find their target, and their venture was a mess. Whole squadrons surrendering to a few British ships in fear they'd stumbled into what was just beyond the horizon the fearsome foe they sought while far from their fellows, and weather doing what it always does in these scenarios. If it all went better, and they had encountered and hamstrung a large portion of the Royal Navy, the British may have felt the need to pull a number of their ships back to protect the home isles. The intention of the Armada was to open Britain to invasion if they could defeat their foes. Such action would greatly better any colony in the West Indies chance of success in throwing off British rule. It really should be noted though that in a confrontation like that, French and Spanish success would've been in no way guaranteed, and the same blow dealt to them by the British could spell disaster for the colonial efforts rather than the British. It may still even if there is no clear victor in the fight if there are few ships left on either side after the battle, which in itself could mean a lack of needed European support for the patriots.
Should the West Indies rebel, keeping them with the United States is a whole further matter. The Thirteen Colonies' greatest concern is their own independence. If French or Spanish liberate British islands that are rebelling, they may just take them for themselves. The Spanish captured the Bahamas and traded them back to Britain for Florida in OTL. Even if these newly spirited patriots did send delegates to the Congress, it may be hard for the Continentals to push for their allies or Britain to allow them their freedom with the rest of the United States since they can't project their power into the area with a few privateers. Or it could be easier, should all three powers feel the need to pull their navies back to Europe to capitalize on the possibility of an invasion, or to defend from it. Even then, with other powers put aside you'll have the States themselves to bypass. Should the West Indies join the Union but the colonies further north not, the Congress is definitely going to be unbalanced between interests (and later between free and slave states), and it still could be if Canada joined honestly (Darn it Quebec).
It could all fall apart in the end, no Constitution agreed upon. I don't believe many of the West Indies have reason to agree with what the more Federally leaning states desired. Though obviously at this point, the proposed intricacies of government could be very different. It could work, though I'm not clear as much to how. It could still work the way it did IOTL, but with several colonies dropping out, possibly with a bit of the West Indies still sticking on.