Searching for the son William's birth/baptism has also been unfruitful. His profile states he was born in Auchencrow in Berwick, which appears to be a small village in the parish of Coldingham. No Clare records appear there before 1800.
There appears to be a lot of "traveling" going on with this family, which makes me suspicious. Christopher was born & died in Staffordshire, England, a wife who was born and died in Ross in Scotland. They had a child in Aberdeenshire, Scotland, who had an "unknown wife" and had a child by her in Berwick. A look at a map, and it makes one ask: sources??