so should I assume his DOB was 1st of May until I can see the original Church record?
Yes.
And was it common for children to be baptised under one name but registered with another?
Yes.
It looks like the family still used Bernard on occasion if it was given on the 1911 census then?
I would say so.
I have a cousin Emby…….. never found it listed in any site.
Her name was Mary Bridget ………MB Emby, she was known as that name all her life.
Maggsie