Don't worry about spellings. My family also came from Ireland with the usual Bryarton spelling variations. Some family members named Bryarton & Brierton are buried in Co. Wicklow in Kilmacanogue and Killoughter. You'll find them on Wicklow Graveyards web. I have most of the headstone transcripts from a Paul Brierton who today still lives not far from those grave sites. The oldest is a Thomas Bryarton died 1804 aged 64. My GG G/F Martin Brierton? arrived in Liverpool early 1800s and two sons born there - Nicholas Briarton 1831 and Cornelius Bryarton 1833. Descendants of Nicholas now spell it Brierton and Cornelius family spell it Brereton.
Other old Brierton/Brereton families lived in and around Dublin, Meath, Laois and surrounding counties.
I have lots more info on this surname, including Scotland, Ireland and elsewhere. Hope I can help.