I think this Lawrence birth is a wild goose chase as Lawrence is too common a surname not to get a coincidence like this.
Births Dec 1902
LAWRENCE Florence Muriel Totnes 5b 121
I've looked at the easier of the two names of the parents - Mercy Marks. The barmaid in Kenilworth (nearby to Leamington) is the strongest candidate by far, but there is no means of proving her connection from tracking just her back through the censuses.
1901 census RG13 2936 folio 36
Coventry Cross Hotel Kenilworth Warwickshire
Mercy Marks 21 Walsall, Staffordshire, Servant Barmaid
1891 census RG12 2251 folio 22
134 Wolverhampton Road Walsall Foreign Staffordshire
William Markes 51 Princes End Head Married Forge Roller
Mary Markes 49 Princes End Wife Married
Daniel Markes 25 Princes End Son Labourer
William Markes 20 Princes End Son Labourer
John Markes 19 Princes End Son Labourer
Selina Markes 17 Princes End Daughter Machinist Stitcher
Sarah Markes 15 Princes End Daughter Machinist
Mary Markes 13 Princes End Daughter
Mercy Markes 11 Princes End Daughter
Hannah Markes 8 Walsall Daughter
Rose Markes 7 Walsall Daughter
Clara Markes 4 Walsall Daughter
plus 1 lodger
1881 census RG112823 folio 134
45 Rollingmill St, Walsall Foreign, Staffordshire
William Markes 40 Sedgley, Staffordshire, Head Married Forge Roller at Iron Works
Mary Markes 39 Sedgley, Staffordshire, Wife Married
Joseph Markes 18 Sedgley, Staffordshire, Son NK
Daniel Markes 16 Sedgley, Staffordshire, Son NK
Cathrine Markes 14 Sedgley, Staffordshire, Daughter General Servant
Harriet Markes 13 Sedgley, Staffordshire, Daughter
William Markes 10 Sedgley, Staffordshire, Son
John Markes 9 Sedgley, Staffordshire, Son
Serlenah Marks 6 Sedgley, Staffordshire, Daughter
Sarrah Marks 5 Sedgley, Staffordshire, Daughter
Mary Marks 3 Sedgley, Staffordshire, Daughter
Muree Marks 1 Sedgley, Staffordshire, Daughter
What was the Erdington address for Arthur's birth and what was Arthur senior's occupation? Was the Leamington address similar to the Kenilworth address in 1901 - a pub/hotel? Have you checked both addresses on trade directories?
Regards
Valda