Tried looking a bit but can't find anything solid - I attached the source for his will / estate / probate file to his Wikitree profile.
Looking at the record, my best guess would be to work backwards, as we have the date of birth in his son's record in multiple places as 8 Oct 1843 in England - if someone more familiar with England records could locate that , and then align that with the other children, it may be a way to find a match.