I'm sorry, I didn't do much Claxton research because he was the stepfather--no blood relation--to my subject Mary Ann Quantrill, son of William and his wife Harriet Buckle.
Have you tried FreeReg and FreeBMD? They can be accessed from the Rootsweb homepage.
Incidentally, while I'm certain my research doesn't connect with William Clarke Quantrill (raider), there is an assertion somewhere on the Internet that his family's Quantrill roots were at Badwell Ash, Suffolk.
Badwell Ash is actually not far from the Tibenham/Old Buckingham area where the Quantrill/Buckle family is first documentable (by me) in 1812.