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I have been trying to find Fullerton connections in Belfast as well.  I'm tied to a Dr. George Franklin Fullerton who moved to Texas following the Civil War https://texashistory.unt.edu/ark:/67531/metapth354360/m1/287/
His Father James C Fullerton may have been born in Belfast circa 1799 and traveled to South Carolina with his family a John F Fullerton born circa 1755 and mother Sarah Edward.
http://www.sharpwriters.com/genealogy/fullerton.html
I've been unable to find any more information if you have ever come across those names in your research it would be much appreciated.
I welcome any Fullerton males to test with familytreedna and join the Fullerton surname project.
https://www.familytreedna.com/groups/fullarton/about

Greetings Wally, I've been trying to track down my Fullerton line in the past couple years. I can only trace my paper trail back to Ireland presumed Belfast, with somewhat certainty. I can trace back to a Dr. George Franklin Fullerton without any real question see citation below pages 277-278. https://texashistory.unt.edu/ark:/67531/metapth354360/m1/287/
He stated his father was a James C Fullerton b. circa 1799 in Ireland, I've come across a site
http://www.sharpwriters.com/genealogy/fullerton.html
stating James's father as a John F Fullerton born circa 1755 in Ireland who married a Sarah Edward born in Antrim County, Ireland. Both deceased in Westminster, South Carolina in 1837 and 1848 respectfully.  From here I'm at a loss, I've seen several Fullerton's listing George Fullerton and Margaret Pirie as being his parents.  Other's have listed a George Fullerton and a Maria Pilborough.  I have since tested my Y dna with familytreedna.com in an effort to help determine the different male lines bearing the Fullerton surname.  There is a Fullerton surname project
https://www.familytreedna.com/groups/fullarton/about
  It sat dormant with no administrator for several years lacking any way for members to communicate, but I have reached out to familytreedna recently, and was able to add a forum for members to post to.  I welcome any all Fullerton's to test with familytreedna and join the project.  Hopefully more light will be shed as more Fullerton's test and become interested in tracing their lines.