Hello Andrea,
I have found a candidate, but it really is difficult to read the name in 1881 & in 1891.
You could convince yourself that it says Byron- or that it doesn't say Byron. Do you have access to the images themselves?
Further detailed research may prove it one way or another!
1881
RG 11/ Piece 2982/ Folio 23
Civil parish: Birmingham
County: Warwickshire
Registration district: Birmingham
Sub-registration district: St Martin
transcribed as
ByrnesHead: John Conley, 68, General labourer, born Ireland
Son-in-law: William Byron (?) 37, House Painter, Born Birmingham
Wife: Anne Byron, 32, Born Ireland
Children:
Maria 12
Kate 10
William 8
Annie 5
Walter 2
George 4 months
......................................................................................................
1891-same family, but only siblings (

)
RG 12/ Piece 2353/ Folio 102
Civil parish: Kings Norton
County: Worcestershire
Registration district: Kings Norton
Sub-registration District: Kings Norton
transcribed as
BynnerHead: William Byron (?) aged 18, Wood worker, born Birmingham
Sister: Maria 22, Housekeeper
Sister: Kate 20, French polisher
Sister: Annie 15, French polisher
..........................................................................................................
I hope that these people turn out to be the right folks.
There seem to be some candidates for the second marriage of Anne Byron.
Perhaps she is living with the 2 youngest children under her new married name.
It's a pity that the father William is not a tailor, and Walter is a little younger than you were expecting.
But perhaps the best bet yet!
UKgirl