Given the oldest kids ages and PoBs, I'd say this was the couple who married in 1887 ... its not the Alfred whose father was Martin, and its not the Arthur we originally looked at !
BUT it has the Bromley / Bow link you mentioned
1901 - RG13/1565; Folio: 77; Page: 12 - Stratford, West Ham
149, Tennyson Rd Alfred Barker 37 Head Commercial Traveller born London
Mary A Barker 34 Wife London
Alfred E Barker 12 son Bow
Laura M Barker 8 dau Bromley
Elsie A Barker 6 dau do
Edward H Barker 3 son Upton Park
Alfred C Barker 5 months Stratford