Martin, thank you for your help. It led to my finding a number of useful things:

On itma I found Old Irish Folk Music and Songs: Part IV Pigot Collection by P W Joyce. (Side note: I have been exploring Joyce. I'm currently reading English as We Speak It in Ireland and A Child's History of Ireland -- I am so thrilled to have found a copy of this 100-year-old book in excellent condition.)

I found an online copy of Kingston Irish Slow Session Tune Book. The site had it at two locations: here and here

Info about the tune of Na Leanbhaí i mBeithil at The Traditional Tune Archive
here

Some biographical information about Liam de Noraidh here

Soon after I started thinking the tune of Na Leanbhaí i mBeithil sounded familiar I discovered that this song is also known as A Leanbh Ghil Mhilis, which I've already sung. I made that discovery here

Well, it's all part of the singing in Irish adventure. I'm sticking with singing Christmas songs till Christmas day, but I confess there will be an element of relief when I go back to simply singing songs in Irish without the additional Christmas song criteria.