I always thought "playing Billy in the bowl" meant playing the song 'Billy in the Bowl' on a whistle/banjo/fiddle etc and winning the girl's heart.

This seems to match with the Pogues lyric in 'the Sickbed of Cuchalainn':"there were lazy drunken bastards singing billy in the bowl".

Nonetheless, i've never come across a song called 'Billy in the Bowl'.