Billy-in-the-bowl is and was a euphemism for sexual intercourse and that is the context in which I understand the line in this song.
In the 17th century a twangman was an archer or bowman, but in the 19th century it meant a pimp and to hawk twang was to be an agent for a prostitute. How this connects with a "twang knife" has eluded my investigation.
Treacle-billy was normally consumed for breakfast by the poor labourers of Dublin it seems that it was a type of doughnut soaked in treacle and was usually vended from handcarts. Carlisle Bridge was, it seems, a favoured place to deal in treacle-billy.

Fergus Russell