I'm only guessing but possibly an "Auld Triangler" is simply someone trying to sing the song "The Auld Triangle". The type of songs mentioned in the article are all much older or specifically about the struggle for freedom and independence of the Irish people. "The old Triangle" is a relatively recent song by, I think, Brendan Behan, and is about, I think, being woken up in prison. Presumably it is considered inappropriate, in the setting described, but as I say, I'm only guessing. I'm sure others will know the truth of it.