I would be interested in seeing the photo that you have, so far i have found 4 J.C (John Campbell) Fullerton ancestors, 2 of which where Lieutenants, although i think there years would be out of place with the one you mention. It would make life easier if they didnt all use the same names!!
My Fullertons seem to orginate from Donegal mainly Ramelton, so its likely there is no or very little connection to the family in Bellaghy but you never know.
Ive got back to -
16 JUL 1696
1766 Rathmelton, Donegal, Ireland
But im not sure il get any further than this Robert.