Hi
I've had a look round and it is quite a hard one to solve

There is a baptism of a Mary Ann PIGGOTT 5 June 1803 St James the Apostle Dover Kent. Parents: John Charles and Elizabeth.
Could be worth trying to track her down to rule her out.
Maybe a simple falling out with his father could be why he changed his name to Pigott, and also putting William as his father.
There are a couple of obits as well
The Sheffield & Rotherham Independent, Saturday, September 14, 1867
PIGOTT ~ Sept. 8th Arthur son of Mr Samuel Pigott of Solly Street aged 18months.
Saturday, April 11, 1868
PIGOTT ~ April 4th Samuel son of Mr Samuel Pigott of Solly Street aged 10 weeks.
claire