Priorto and during the revolution the Bahamas did in fact had a fair amount of rebel sentiment (Loyalists migrated there after the ARW was over)
1. You would first need to increase the amount of rebel sympathy however, by increasing it overall. or doing something more specific to anger the Bahamians. (possibly Britain siding more favorably with Bermuda in the Bahama - Bermuda dispute over salt rights on the Turks and Caicos islands)
2. Get someone from the Bahamas to at least go to Philadelphia, as a representative of some sort. Petitions were sent to all the colonies for joining, inlcuding the ones that didn't (like West Florida, Bahamas, etc.) Formally joining the 13 colonies might not explicitly happen though even if the representative they send is just an 'observer' or someone who did so on their own initiative (but should have at least some level of support and/or influence)
3. Realize that, as an island colony, going against the powerful British navy was the main reason they declined, even if they largely looked the other way when America ships did arrive and raid their gunpower stores. They can though be more belligerent and uncooperative with the British in your TL. Give Britain enough frustrations with the place that they care much less about keeping it. (perhaps make them act for want of bringing in wealth as an illegal freeport, smuggling center, to a greater degree, but be aware of the consequences and Britain's reaction to this.)
4. Also realize that ultimately, the US will have to obtain it from France or Spain. Only they have a Navy strong enough to take and hold it to the end of the war. Spain did so in OTL, but gave it back to Britain for the Floridas IIRC. I believe France had an agreement with the US that if they seized Bermuda they would give it to the US at war's end, they may also be true of the Bahamas, but I can't recall. Spain had no such agreements.