When looking at the Bahamas, you must look at the history of its neighbors as well. There were six British territories in the Caribbean, and was arguably more similar to the southernmost of the thirteen colonies than they were with New England. Plantations made up the basis of the economy, with slave-owning being commonplace among the more wealthy plantation owners. They developed in a similar way to the colonies in the modern day United States, and were ruled in a similar salutary neglect manner. However, a major point of divergence was the Stamp Act in 1765 -- as you may already know, there was major opposition in the modern United States. Violent opposition was less common in the Caribbean, and discontent was more common among British-based landlords. This does not mean it necessarily went over well; Antigua dodged it, St. Kitt's rioted, and Jamaicans issued threats towards officials. The Bahamas in particular did not have any notable unrest.

The coercive acts, the result of unrest in America, did not punish Caribbean colonies, and public outcry was therefore limited. When the American War of Independence started, the only major church present on the Caribbean colonies was the Anglican Church, perhaps contributing to a lack of support for the American revolution (1). White landowners, the ruling class of the islands, did not side with their counterparts in America. A major reason for this was the legislation passed by the Continental Congress, which would have caused economic crisis in the Caribbean (2). There was no chance, after 1774, that the Bahamas or any of the other six Caribbean colonies would have joined the revolution. The Bahamas during the war played a relatively minor role - Nassau was captured twice. First by American revolutionaries, by Commodore Hopkins, and second by the Viscount Galveston, a Spanish general. The Bahamas were returned to British control after the end of the war.

References

1 - https://www.c-span.org/video/?323046-1/discussion-american-revolution-caribbean

2 - An Empire Divided: Revolution and the British Caribbean (where I started on this topic. Heavily recommend you read it to go further on this)