Where you a Bahamian in the mid to late 18th century perhaps? The attitudes of modern Bahamians, separated by 250 years or so and the influx of loyalists and later those freed from slavers by the Royal Navy, aren't really reflective of the political situation then.I can't speak for Bermuda, but no way, absolutely not, do the Bahamas want to go to the US. They might get signed away in a treaty, but the population is going to want to stay British. I know. I am Bahamian.
I think this whole discussion hinges on whether they become a state. If they are territories they are simply given away at the end of the War o '12, but if they are a state, they are integral US territory, and even in the Nappoleanic era the world is not a game of risk. The US isn't going to sign a treaty giving away part of it's core metropolitan area, occupied or not.
To put it in modern terms this wouldn't be like signing over Guam, it would be like giving up Arizona. Any treaty that does it unacceptable to Congress, which in turn probably leads to longer War of 1812 if the British give enough of a damn to keep fighting over them with a much harsher peace imposed.