Although not exactly close to Carrot Bay, one of my favorite legitimate local places is the Midtown on Main Street. It's truly a local joint, although the proprietor, Gloria, is no amateur. Homemade everything - local fish, conch stew, souse, goat water soup,roti - you name it (depending on the day, of course).
And the piece de resistance - locally made ginger beer and mauby.
Gloria is quite the colorful character, and an old friend (please give her a hug for me).
As an historical anecdote, the basement of the Midtown used to be a 'dive' bar, that the cast of the movie 'The Deep' discovered, and frequented during filming. It used to have a huge shark head hanging over the entrance. Of course the shark and the bar are long gone. It's just used for storage now.