The reg above should be enough for the Walsall office to find it, it does not say were born only the reg district.
I would have a chat tell them his birthday and father is defo Arthur, if these details are correct you know you have the right birth cert for your grandad.
This looks likely.
1901 - RG13/2699/100/59
64 Croft Street, Walsall Staffs
Arthur Dobson, 36, hd, m, ironworker, Walsall
Sarah, wife, 28 Walsall
Sarah A, dau, 2 "
Samuel, son, 9 "
James A, son 3mths "
John Tiff 27, s, bro in law ironworks/pudler walsall
Marriages: Arthur Dobson - Sarah Tift Jun Q 1898 Walsall 6b 1292
Could be second marriage with Samuel age 9? But if these are your folks looks like Sarah would be the mother of Harry.
Cas