So far as I know the song that Tom Lehrer calls "The Irish Song" or "The Irish Ballad" is the song that is popularly known as "Rickety tickety tin" When he introduces it on the tape that I have, he explains that Irish songs all seem to be bloody and have too many verses, so he thought he'd try his hand at one. If we need accuracy, I could go listen to Mr Lehrer on the tape.
I have no idea whether I heard CMT sing this or not.
But what's in the song book is the words for "rickety tickety tin".

Blessings,
Barbara