Hi
The last child that I can find was b. 1787:
1774 - James
1776 - Archibald
1778 - Joseph
1780 - Thomas
1782 - Elizabeth
1784 - Archibald
1785 - Margaret
1787 - David
I've checked baptisms for 1740 to 1758 for a Marg P*rv*s and there are two possibles if a naming convention was used and you're sure Margaret was b. Coldingham
9 Sept 1748 - Father - James Purvess
15 May 1743 - Father - David Purvess
both Coldingham
No mother's names given.
I've checked marriages in Coldingham for these two possible fathers:
6 March 1739 David to Elisabeth Landells
24 July 1742 David to Margaret Purvess
30 July 1734 James Purvess to Elisabeth Purvess
Not sure if that narrows things down or not but at least some possibles

Nothing in Cockburnspath where most of the children where baptised.
Info from SP as the IGI were submitted entries.
Maybe MIs (Monumemtal Inscriptions) might help narrow things down.
How sure are you that Archibald Davidson was b. 1715? I don't see any bpt. Berwickshire 1714-1759 using soundex although there are some bpt. in other counties.
Gadget