I realise you have most of this and I note the other records you mention on Ancestry. It looks likely that the naming pattern has not been followed and the use of Alison as a middle name for Patrick in 1811, together with the 1756 marriage of David and Dorothy seem plausible but I do not see any baptisms for any children.
I have posted below in case it helps others.
Records on freereg unless stated.
Thomas Frater a shepherd & Isabella Weddle had the following children baptised at Kirknewton
Abode Kilham, then Akeld (Mothers surname on baptisms)
Thomas FRATER 8 Sep 1797, George FRATER 22 Dec 1798, Mary FRATER 23 Jun 1800, Jane FRATER 23 Apr 1802
James FRATER 24 Mar 1804, John FRATER 10 Jan 1806, David FRATER 12 Sep 1807, Isabella FRATER 11 May 1809
Patrick Alison FRATER 2 Jul 1811, Mathew FRATER 19 Aug 1813, Robert Allen FRATER 11 Feb 1816
Burials:
Dorothy FRATER Burial 13 May 1804 Northumberland Bamburgh: St Aidan: Bishop's Transcript
Notes: Died 11 May 1804 age 86, Widow, abode Newham. (Born about 1718)
Two possible burials for a David?
David FRATER Burial 06 Dec 1799 Northumberland Lowick: St John the Baptist: Bishop's Transcript
Notes: Died 3 Dec. Late a Shepherd age 70 abode Holburn Grange (born about 1729)
David FRATER Burial 27 Mar 1779 Northumberland Kirknewton: St Gregory the Great: Bishop's Transcript
No age abode Fleehope.
Burial one maybe, I will not repeat the words used on the notes but basically he was illegitimate father David.
David FRATER Baptism Mar 1728/9 Northumberland Doddington: St Mary and St Michael: Other Transcript
This seems to be the only record in the County for a David on freereg before 1750.
Thomas 1769 – 17 April 1839 Bamburgh on find a grave.
Isabella 1771 – 7 July 1858 Bamburgh on find a grave.
In 1851 Isabella age 79 (1772), widow of a shepherd on parish relief (transcribed born Kirknewton but census form not clear eg: Ord?) is in Bamburgh with her daughter Isabella and her daughter Jane.
Colin