freddam,
I assume this is the James you refer to (you haven’t give any dates):
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1853/09443/5420815.pdf
If so, James was born c 1831 and lived in Ballylesson at the time of his marriage.
Possible sibling: John Sutter in Ballee, parish of Connor:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1850/09382/5397947.pdf
And Robert Sutter:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1850/09382/5397953.pdf
And Rose Sutter in 1851:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1851/09402/5405733.pdf
Weavers and labourers moved about a lot and can be difficult to trace. I can't find a death for Bernard Sutter 1864 onwards (which is when death registration began) so suspect he died before that. Unless he was Church of Ireland, there are unlikely to be any burial records. Nor can I see gravestone but that wouldn’t be unusual for a weaver as most could not afford one.
Tradition was to marry in the bride’s church. Rose married in Ballymena 1st Presbyterian so that suggests it was the family church (as you have already said). Their records start in 1812. There’s a copy in PRONI in Belfast. You might find some of the family recorded there.
From his marriage certificate, we know James was born around 1831. That suggests his father Bernard was born around 1811 or earlier. Sadly few churches in the Ballymena area have any records for that period. (The Church of Ireland has records from 1789 but there are a number of significant gaps). So probably there’s no record of his baptism. His marriage would likely have been in his wife’s church. Do you have any idea where that might have been?
Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.
The North of Ireland Family History Society is running an Ulster DNA project in conjunction with FTDNA and can offer testing kits at a reduced price. http://www.nifhs.org (Go to DNA project on the website).