I was talking to Shirley, a granddaughter of William Barfield. She told me this story: William Barfield died at home. His family built a coffin for him, a simple wooden box. They took him down to the Hawhammock Church Cemetery on a horse drawn wagon and had his funeral. They didn't hire a mortician or funeral home, but did things the old fashioned way. It appears that the family either didn't know about the recently instituted laws requiring death certificates, or else they didn't care.
So there's no death certificate; I can stop looking. Unfortunately, I'm no closer to finding William's father.