It does seem rather a tall order, Fay, but many studies begin that way.
I believe that many , maybe most, of the street singers in London and also those on stage and in taverns did include inuendo and specific references to matters outside the text, as you say. References to previous performances by themselves and others were part of their act. There was also "class-based" comedy which has always been done so well by the Brits. That would include cryptic messages and insults.
Whether you can find actual documentation, from the period you are looking at,is the thing. The broadsides will give some versions of songs and may show adaptations and variants from particular performers. I would agree with McGrath that Blackface and Musichall didn't seem to show much overlap in those early days.
Maybe,(thinks I on the run), street performers are the key? There is a great variety of acts among them. They didn't all take up Blackface. Finding a niche was necessary for survival. They had to balance familiarity and novelty to survive. There could be remnants of the style you are seeking among them. "London Labour and the London Poor" by Henry Mayhew may help.
His interviews with street performers came just after Blackface became popular but their may be clues there. Cheers, Joy