I take it he's this John Bate in Oldbury in 1851:
HO107; Piece: 2024; Folio: 542; Page: 6
If you want confirmation on Phoebe's maiden name you will need James Bate's birth cert.
I can't see them in 1841 but in 51 he says he's 50, ok I know ages can be out a few years but have you discounted this John:
John Bate
Ch.1 September 1799
SAINT THOMAS,DUDLEY
Parents:
Isaac & Elizabeth Bate.
There's also this one;
John Bate
23 January 1803
SAINT THOMAS,DUDLEY
Parents:
Joseph & Mary Bate
Also have you discounted this marriage:
John Bate
Spouse's Name: Phoebe Foxall
Event Date: 31 Jan 1820
Event Place: St. John, Halesowen,
Details:
St. John the Baptist Halesowen
31 Jan 1820
John BATE / Otp / Bachelor
Phoebe FOXALL / Otp / Spinster
Wit. Richard FOXALL Rebekah BATE