A Baptism for Arthur Edward Banks 14/4/1876; parents Frederick & Sarah; Jeweller.
Add: Heath ? St.
You can see the originals without having to but the certificates.
I'm afraid this isn't quite the case.
Baptisms aren't the same as the original Birth certs.
A child may have it's birth registered but not necessarily be Baptised.Also a Baptismal record won't give mother's maiden name & usually doesn't give DOB.
I'm afraid there's no substitute for a birth cert. to establish who someone is.
These are all the Banks marriages I can find with Frederick, jeweller as father:
Frederick b.1854 m. 1888 father listed deceased.
Samuel b.1859 m.1882
Florence b.1864 wit: Frederick Sidney Banks & Emma Banks
Emma b.1865 m. 1887 father listed deceased.
William b.1866 m.1888
Annie b.1867 m.1887 father listed deceased.
Looks like the children of Sarah all believe their father is dead.
There's only 1 Frederick Banks in B'ham & that's the one you've found.
Trade Directories;
1858; 83 1/2 Carver St. home address given as 7 Morton St.
1862-66; 76 Carver St.
1868; 149 Warstone Lane. Here until 1905.
Variously listed as Jeweller, Goldsmith & Jeweller, Jeweller & Silversmith & Manufacturing Jeweller.
1908: Frederick Jnr. has the business.