None of your "peeps" in my tree, but I have been digging about for Barnet. I found him in a couple of the censuses.
If I am correct, he was married more than once.
I can't find any record of Barnet's birth, but I am going to make a suggestion to you as to why you haven't found his birth as yet.
My theory is that Barnet was a middle Christian name and that he was baptised under his first Christian name. I would go even further and say that my suspicion is that Barnet was his mother's maiden name (quite a common trait in those days to have mother's maiden name as a middle name)
It may well be worth looking for a marriage between Fullard/Barnet - this might throw up Barnet's parents.