If memory serves me correctly the Bahamas were occupied by Spain during the ARW and in the post war was in one of those places that if the Americans pushed for it, the British may have conceded it.
Of course come War of 1812, it may be taken back and the US would almost certainly view the war as a loss (its considered a tie or victory by most Americans)
The Bahamas was occupied by the Spanish, for one year, [1782-3] during the ARW. The Spanish garrison in Nassau surrendered to a small force of American loyalists and Bahamian millitia, led by Col. Andrew Deveaux, in 1783. In the years after the war, thousands of loyalists, including some of my ancestors, came from the mainland. Most didn't stay here for too long, but they were a potent political force while they were here, and would have been very opposed to any sort of political union with the 'rebels'.
The Bahamas are generally pro-American, but I can't think of any POD where the British would be willing to give them up. Maybe a victorious CSA, with Britain as its ally, might come to dominate the area, and as is sometimes the case with Cuba in CSA victory TLs, could maybe adopt the islands as another state, but not much here then, pop. 50k max. at that time.