A sugar-baker was the owner of a sugar house, a factory for the refining of raw sugar from the Barbados. Sugar refining would normally be combined with sugar trading, which was a lucrative business. The architectural historian Kerry Downes gives an example of one sugar baker's house in Liverpool being estimated to bring in £40,000 a year in trade from the Barbados.
Sugar houses are factories for refining raw sugar, specifically from the country whose capital is Bridgetown.
This is correct, Bridgetown is the capital of Barbados.