I know this is a late reply but I lately noticed a Hawkshaw-Basquill marriage where Basquill was spelt Baskwell - on church records (which includes marriages - these are recognised by the state) it was up to the priest/minister how the name was recorded. I often see a different name variant of the baptism and the civil birth registration. Might be a small help.
Oh hey, this is my Hawkshaw line. Nicolas Basquil married Eliza Hawkshaw 1861 in Ayle. They migrated to Jackson County, Ohio shortly after the wedding and had a passel of kids. Nicolas was b. 1816 in Mayo so may be a brother to your Patrick?
There's also a Michael Basquill married to an Ellen O'Brien in Islandeady parish of Tuam (which covers Ayle) in 1874. He's from Brushbrook and father is Michael Basquil, landholder.
There's a Walter Basquil, also father Michael Basquil landholder, who resided in Aughagower at the time of his marriage in 1871.
A Margaret Basquill, 23, married in 1871 gives her father's name as James Basquill, Landholder (still alive) they were from Killavalla, married in Castlebar.
Going back further, I see a William Basquill marrying in 1868 from Tubberane (also spelled Tubberuane) father Michael Basquill carpenter. Also Castlebar.
