If Bath haven't done so, you could order from the GRO by using the 'reference not known, option to check what parents names have been given.
Once you enter the year, you then have the option to request a cert for Henry Fido (because that's what the name will be, but you can specify the parents names as George Fido, put the mother down as "Hannah or Anna", both names in the box and her maiden surname.
If there isn't a match, they'll send you a refund.
It's an interesting dilemma if this does turn out to be the child you are looking for.