I don't know if there is a connection or not....
I am one of many Sullivan descendants from Cappaphaudeen. Two other family members and myself have only been able to trace back to Patrick O'SULLIVAN and wife Ellen DEMPSEY. Patrick was possibly b. abt. 1834 Cappaphaudeen, Co. Cork, and Ellen b. abt. 1836 Co. Kerry. One of their 8 children was Patrick Peter SULLIVAN b. Cappaphaudeen, and his God parents were Michael SULLIVAN and Ann CARROLL (we have conflicting b. date info for Peter so it's fr 1872-1874). Maybe your Michael is a brother of my Patrick? All but one of those 8 siblings left Ireland for America, a few descendants still live on the Ireland property. CARROLL's and NOLAN's are also in my tree, however my Nolan's are not the same generation as yours, ours are further down starting when Nora SULLIVAN b. 1936 married Michael NOLAN b. 1925.
Location/District, etc. have included:
Rockchapel, Newmarket, Cappaphaudeen (alternative spellings: Cappaphadeen, Cappaphandeen, Cappafadeen, Cappafadine, Cappafadedeen, Paddy's Plot ), Toorreenfineen, Kanturk, Brosna, Clonfert, Clonfert West, Kilmeen, in Co. Cork and Co. Kerry. Real close to those borderlines.
Other additional family names: MURPHY and GLEESON (and suspect others).