Could you give a little more detail about your interest in this family. Some first names and dates would be helpful. Are you talking about the William BEARD who discovered the Banwell Bone Caves in about 1824?
Have you been able to find a christening for the William BEARD who was born in 1805?
Have you discounted the christening in Banwell in 1802 of a William BEARD? As you have his death date of 29th April 1871 it does look - from the 1871 census entry at least - that he might be the one married to a Betsy and whose birth date is given as 1802 on the census but perhaps this isn't the one you are looking for. However, the Death Registration on FreeBMD for the June Qrt 1871 does give his age as 69
BEARD William 69 Axbridge 5c 404
The William BEARD who was so closely connected to the Banwell Bone Caves was born in 1772 and died in 1868 but it is possible that he might have been related - possibly a half brother to the father of your William BEARD.
I think all the clues are in the parish registers for Banwell which are online on this site.