I have not heard of a Missouri bar fight tied to the Fultons. Do you have a time frame?
Adelia Fulton had a granduncle Horatio Fulton who left town and was never heard of again. He was brother of Robert Jefferson Fulton, the murder, NOT the Horatio P. brother of Robert Nero who died in the Civil War. It seems unlikely this would be who the story referred to but you never know.
Seeing as Adelia died young it would seem probable that if a story carried through on her line it was probably more contempory. I don't have knowledge of any of the Fulton descendants in Missouri but Adelia's older brother Clarence lived in N & S Dakota with his Norwegian wife and 2nd family. Some of these children, who would have been first cousins of your grandfather did end up in Michigan, Washington and California.
Let me know if you have more info as to name, time frame, etc.