Thanks so much for the reply, JONMONASH.
When your original ancestor came to Tullynaskeagh, Clan Cameron were not covenanters. In fact, they fought alongside Clan Donald against the Covenanter Campbells (Eventually, the Campbells won out and took over Kintyre). Clan Cameron remained catholic and later fought on the Jacobite side. That being said, no clan was fully homogeneous in religion and your family may have taken the pledge. In fact, that may explain their emigration to Ireland because, at that time, they would have been shunned.
I have studied the yDNA of Clan Cameron and the vast majority are R1b. I have found only one R1a Chalmers. So, I originally discounted Clan Cameron as the originators of the County Mayo Chambers. However, if the Tullynaskeagh Chambers are R1a and are Clan Cameron, then I have found what I'm searching for. Contrary to Padraig Mac Giolla Domhnaigh's "Some Anglicised Surnames in Ireland (1923)" Chambers in Antrim/Down before the Plantation did not originate from McCambridge of Kintyre.
To your knowledge, has any Chambers male in your family taken the yDNA test?
Thanks again,
Tim Chambers
PS: I'm hoping that our families are related. That may explain the mechanical skills that saved the family from famine and ensured success in Liverpool and America.