I thought this is a chance to find the few possible locations in Ireland. John
Grenham ‘s website started my search well,
https://www.johngrenham.com giving several catholic marriages new records and he lists alternative spellings as Barrens and Barrins
I then tried using one source Tithe Applotment Books 1823-1837
to locate the possible place james Barrons might have come from
1.Tithe Applotment Books 1823-1837 Part 1 in the 26 counties
http://titheapplotmentbooks.nationalarchives.ie/search/tab/index.jsp Barren has 53 entries
Barron has 243 entries
Barrin non entries
Barrens Henry Crumlin Kilmore Monaghan 1825
Barrens Thomas Denover Moynalty Meath 1827
Barrens Thos Denover Moynalty Meath 1827
Barrens Thomas Shanearn Moynalty Meath 1827
Barrens Baltrasna Moynalty Meath 1827
Barrens Patrick Shelbagin St James and Dunbrody Wexford1833
Barrins none
Barrons Shanakill St Patrick's Limerick, Clare 1825
2. Tithe Applotment Books 1823-1837 Part 2 in the 6 counties originals in PRONI, Belfast
my index of the Irish Gemealogy Hub listing by parish see
http://www.irishgenealogyhub.com/Barren, Charles-Townland: Derrahollow Year: 1832-Belleek-Fermanagh
Barren, Dr. Townland: Edenderry Year: 1828-Seapatrick-Down
Barren, James-T: Unagh Y: 1827-Lissan-Derry & Lissan
Barren, James-T: Unagh Y: 1827-Lissan-Tyrone
Barrens, Henry-Townland: Derryarnish Year: 1834-Aghagallon-Antrim
Barrens, John-Townland: Moyntaghs Year: 1834-Aghagallon-Antrim
Barrens, Robt.-Townland: Ballycavin Year: 1834-Aghagallon-Antrim
Barrens, Widow Townland: Deerpark Year: 1834--Glenavy-Antrim
Barrens, William-Townland: Derryarnish Year: 1834-Aghagallon-Antrim
Barrin/Barrins no entries
Barron, Andw.-- Townland: Divernagh (Whaley) Year: 1835-Killevy-Armagh
Barron, Andw.-- Townland: Divernagh (Whaley) Year: 1835-Killyman-Armagh
Barron, Charles-- Townland: Divernagh (Whaley) Year: 1835-Killevy-Armagh
Barron, Charles-- Townland: Divernagh (Whaley) Year: 1835-Killyman-Armagh
Barron, Chas.-- Townland: Divernagh (Whaley) Year: 1835-Killevy-Armagh
Barron, Chas.-- Townland: Divernagh (Whaley) Year: 1835-Killyman-Armagh
Barron, J. Counselor-Townland: Kinghill Year: 1834- Clonduff-Down
Barron, James-- Townland: Divernagh (Whaley) Year: 1835-Killevy-Armagh
Barron, James-- Townland: Divernagh (Whaley) Year: 1835-Killyman-Armagh
Barron, James-T: Stughanbeg Y: 1826-Tullyniskan-Tyrone
Barron, James-Townland: Carmavey Year: 1827-Killead-Antrim
Barron, Jas., Jr.-Townland: Loughmorne Year: 1816-Carrickfergus-Antrim
Barron, Jno.-Townland: Loughmorne Year: 1816-Carrickfergus-Antrim
Barron, John-Townland: Ballylinney Year: 1835- Ballylinny-Antrim
Barron, John-Townland: Ballylinny Year: 1835- Carnmoney-Antrim
Barron, John-Townland: Carneil Year: 1830-Raloo-Antrim
Barron, John-Townland: Carneil Year: 1830-Ramoan-Antrim
Barron, John-Townland: Ednigo Year: 1834-Dromore-Down
Barron, John-Townland: Lisnaward Year: 1834-Dromore-Down
Barron, John-Townland: Loughmorne Year: 1816-Carrickfergus-Antrim
Barron, John-Townland: Monkstown Year: 1835- Carnmoney-Antrim
Barron, John-Townland: Slunmerow Year: 1816-Carrickfergus-Antrim
Barron, John-Townland: Toneylisderritt Year: 1834-Cleenish-Fermanagh
to be continued