Are you after local cuisine or a place where locals eat? I would think Negril, Jamaica might be better, no car required as you can stay on Seven Mile Beach and there are plenty of restaurants serving up local cuisine.

The St. Lawrence Gap area would work for you.

Barbados would be a good option for getting out and about and enjoying the island and its cuisine, however to truly go to those places that serve local fare that are also where locals eat it car is recommended at least for a couple days. That way you can experience really fun local limes and eateries not necessarily in an extremely convenient areas.

St.Lawrence gap on the south coast, is a great area to stay in with all the conveniences and amenities close by and on a great bus route connecting you to the Capital of Bridgetown. It's the most happening spot where locals and tourists both hang out, plus there is a street vendor who operates a mean grill and serves up Bajan food and there is also a restaurant in the Gap called sweet potatoes.

However some great weekend limes and lunches happen at places like lemon arbor in St. John ( saturday afternoon), Oistins fish fry ( Friday/Saturday nights or any night) in Christ Church - not far from St. Lawrence Gap, and the Atlantis hotel on the east coast has a sunday buffet Bajan spread not only is it value for money but the food is really good.

Hope this helps.