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