I can't offer too much on this except that the song is not Master McGrath although it may be written to the same air. The name of the town in verse two is not McCroom but Macroom (máigh cromtha in Gaelic) which is in County Cork, sounds like the dog was bred there and brought to County Clare later.

I have heard songs about greyhounds that come from the area of Baile Mhuinre (Ballyvourney) which is a Gaelic speaking area quite near Macroom. One of these was called "The Big Black Bitch from Clare" which to my memory was about a dog rather than a person. I don't think this title refers to the song you're looking for, but I hope this limited information might assist you in your searches.