Here's some interesting but possibly useless information:

This site, this site and this site mention a song called, "You Will Have to Sing an Irish Song (Where the Little Shamrock Grows)," sung by Nora Bayes in "Follies of 1908," which was the second of Florenz Ziegfeld's Follies.

This site and this site mention a song called "It Takes a Great Big Irish Heart To Sing an Irish Song," by Al Herman & Jack Glogau, 1915.

This site mentions a song called "Sing an Irish Song," recorded by The Barleycorn on "My Last Farewell."