The Prince of Wales in question was Prince Edward, who would later become King Edward VIII. Born in 1894, Prince Edward was socially very active during the 1920s, and was a much sought-after member of aristocratic society.
In 1936, he provoked a constitutional crisis when he decided to marry the American divorcée Wallis Simpson; this decision led ultimately to his abdication.
Listen to the abdication speech of Edward Vlll:
There is no such club as Bratts, but there is a gentlemen's club in London known as Pratts. It was founded in 1857, and is situated off St James's Street in Mayfair.