First of all, Britain would never have allowed any of its Carribbean islands to become independent. At the time the US gained its independence in 1783, the West Indies were far more valuable to the empire (commercially) then the former colonies. In fact, Jamaica alone generated more revenue for the Crown than all the former colonies combined.
Secondly, the USA, even with French assistance, would not be in a position to threaten to destabilise Britain's presence in the West Indies. For the obvious reason that the Royal Navy dominated the Carribbean and the Americans weren't able pose any significant martime challenge for another century.
Finally, the social structure on Britain's West Indian islands was very little like colonial North America. The plantation owners who controlled the islands were heavily dependent on British protection and the markets for their sugar which the empire provided. Also, many of the influential figures in Parliament owed their prosperity to the sugar plantations of the West Indies, and would have acted strongly if they were in any danger.