"We also Know that a visiting floorsinger was laughed at for singing the Rock Island Line with a cockney accent."

I would probably have laughed as well (or sighed heavily)! In the post-war period, guitar-based American style music was becoming more and more popular in the UK. I would guess that there were, at that time, lots of singers, performing that type of music, who were looking for a platform. A club in central London, that encouraged singers from the floor, represented an ideal platform for them. I would guess that part of the reason for the Singers Club's proscriptive policy was to avoid being swamped by Cockney pseudo-Yanks!