IT is highly probable that the families are related, but it may take a litte effort to find the exact connection. I will explain my sources:
I was given a lot of family tree information by a nun cousin, a Byrne descendant, who in turn had been given a handwritten family tree by a cousin of hers, one of the Byrnes at Bansha, probably about 1980.
He wrote of a family descended from Edmund Byrnes and Catherine Woodlock with at least 7 children born about 1840-1850. 4 of the children- THomas, William, Jack (presumably "James"), and a daughter all went to the US. Denis stayed in Ireland and is the ancestor of the current Bansha Byrnes.
While this source did get some names and dates wrong, is it possible that you are descended from the "Jack" (or James) above?
I am descended from another line which went to Ballylusky, near Killenaule in NE Tipperary in the 1870's.
Feel free to reply directly to my EMail address which should be in this message.