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The licence says that the bride is a minor and her mother (a widow) has given consent for the marriage, as the father is dead and there was no other guardian.

This John Beckwith marriage looks promising.
https://www.familysearch.org/ark:/61903/1:1:NFCQ-9YQ?from=lynx1UIV8&treeref=MVQK-7J1

Cawthorne is only a couple of miles from West Bretton.

Some time after the death of Barnabus it's said his widow married again

Alice May PRATT birth registration
PRATT, ALICE  MAY     Mother’s name RYE 
GRO Reference: 1887  D Quarter in WESTBOURNE  Volume 02B  Page 401

Birth date 13 Oct 1887
Parents: Henry PRATT and Alice RYE who were married 1875 Froxfield, Hampshire

1901 census – Alice PRATT
https://www.familysearch.org/ark:/61903/1:1:XSNG-WKS

1891 census - Pratt family
https://www.familysearch.org/ark:/61903/1:1:43WK-P3Z
1881 census  - Pratt family
https://www.familysearch.org/ark:/61903/1:1:Q27H-M99N

Alice Mary/May PRATT - 1st Marriage 1905 Westbourne, Sussex to Barnabas JONES

Alice M (PRATT) JONES – 2nd marriage 1917 – Alverstoke, Hampshire, to Joseph William VOWDEN (born 15 Mar 1895 South Stoneham)
More children born to Alice and 2nd husband (including a son named Gerald Bertram VOWDEN born 1925, who died in the Navy during WW2

Alice May VOWDEN death 1978, New Forest, Hampshire, age 90

Hi Genni
You have done well to research the Hugh Kerr - Marg Lochead birth and marriage and eliminated him / them.

You may know that there are a couple of other births that you should probably research for elimination or not.
They are both in Leith South (not Ratho, but fairly close)

KEIR   PETER          PETER KEIR/MARGARET DAVIDSON FR2709 (FR2709)   M   
21/08/1817   692/2    80 449   Leith South


KEIR   PETER          PETER KEIR/MARGARET SMITH FR2735 (FR2735)   M   
24/11/1818   692/2    80 498   Leith South

It is quite possible that there is no birth record to be found for your ancestor. It may be one of the many records that never existed or was lost.
The fact that there is no parents named on his death certificate might be an indication that he was illegitimate. If this is the case, it might be that Kerr was Peter’s mother’s surname.

My only other suggestion would be to look in the Ratho kirk sessions around the time you are interested in, to see if there is any mention of the Kerr/ Keir name. The session minutes can be very useful in picking up illegitimate births. These are on Scotlands People here ....
https://www.scotlandspeople.gov.uk/virtual-volumes/record-creator-search/sessions-volumes?rex_uid=REX00327&related_places=GAZ00936&placename=Ratho

This is the picture of building site that was posted on another thread. You can see that it is a sizeable building, when you see how small the workmen are in comparison.