Hi Denis, This morning I went through the entire 19th Cen. church Register of Bermuda book compiled from all the church registers on the island (not catholic mind you) and I looked under: Baptisms of BEDFORD - nothing at all. Baptisms of child born under a different surname to its father - nothing found. Baptisms of NO NAME children (illegitimate) - nothing found.
The only suggestion I have for you is to contact the Registrar General, Government Administration Bldg. 30 Parliament Street, Hamilton HM12, Bermuda. Tele: 441-297-7693
Because he was born prior to birth registrations (which began abt. 1866)and he may not have been baptised on the island,it is possible you will not find anything.However, it could be that his baptism record was missed and not recorded in Hallett's book.(some were missed so I am told). Also, if you think there is a possibility that he was a child of one of the various UK Regiments here on the island, some of the birth records are housed at the Portsea Hall of Records in Portsea, England. This told to me some while ago.I have never requested anything from them as yet.
Also, in the census you found in 1881, did not mention a child of his or a wife or his profession.Is there not a clue somewhere there you could go on.Sometimes we have to go east or west in order to go north. Good luck in your search. If I can be of further help just contact me. Judy