Parish registers for Barton were published in The registers of the parish church of Barton, Westmorland. Baptisms and marriages. 1666-1812. Burials, 1666-1830 (1917) by Henry Brierley (online here), and contain only three people named Barton, plus one who has Barton as his middle name:

"An w. of Tho: Barton" 8 November 1684 [Burials p71]
"Anne Barton of Sockbridge Mill, Spinster" 27 April 1717 [Burials p78]
"George Barton s. of Eliz: Todd Widow (late Scholough)" 25 February 1791 [Baptisms p49]