Hi Brooke,
I searched on the site rootsireland.ie for William Brannen in the 1700's in Mayo and Kilkenny but had no hits.
I then tried all counties and got William Branan baptised in County Wicklow in 1726.
There weren't any other William Brannens in the later 1700's so maybe this is the one who emigrated.
With poor literacy and the spelling of names changing over time, don't be too worried that the spelling isn't a perfect match.
On ancestry there are lots of family trees showing William Brannen who .
His birth varies from 1745 to 1760 but he married Elizabeth Gross in 1780 in North Carolina and they had 12 children.
His parents were Thomas Alexander Brannen and Susannah Terrell both born 1730.
I could not verify any of the ancestry information on rootsireland, familysearch or findmypast.
ONn PRONI (Public Records of Northern Ireland) there is a freeholders list and John and Susan Brannen are mentioned in 1815 in Armagh.
I'll think more about it but it looks fairly certain that he was from Northern Ireland.
Best regards
Col Cafferky