Marriages usually took place in the bride's parish if different from her husbands, first children could also have been baptised there if an RC family. Barton sounds Anglo-Irish.
I'm not sure you can go back into the last 1700's in Fermanagh. There are no records for the Tithe Applotment in Fermanagh 1823-1837.
Griffith Valuation 1860 is the next closest census substitute quite a big gap. www.askaboutireland.ie
Ederny, Fermanagh, was a Town and a Townland in the Parish of Magheraculmoney, in the Poor Law Union of Irvinestown
There are 48 Barton's listed and 1 Deazly in the Parish of Magheraculmoney.
If James is the oldest son his father was probably James but he did name his oldest Robert John which is probably his dad's name
There are 3 James Barton's in the Griffith Valuation in the Parish of Drumkeeran but unlikely to be the father because of the age. If James was born 1785 then he father would of been born at least 20-25 years before that.I don't see any Church of Ireland records earlier than 1801 and Magheraculmoney or Drumkeeran are not even listed
There were 33 Barton's living in the Parish of Drumkeeran
some owning their own land, any family stories?
I did look up Tubrid CofI and it is Drumkeeran rather than the above. There is a picture on flicker of the church. You can google Tubrid Anglican Church-Ireland
There are 3 Barton's in the current British phone book for Barton's in Fermanagh and 0 Deazly.