There is no evidence in the story that the lynch mob actually included members of the Ku Klux Klan. The author never describes them as such, and the men--several identified as Cunninghams--wore regular clothing rather than their white hoods and robes. They made no attempt to hide who they were, and their courage to undertake such a criminal act was probably aided by alcohol, rather than by a clandestine appearance. Atticus actually tells the children beforehand that
"The Ku Klux's gone... It'll never come back."
Atticus must have been trying to calm his children, who were upset by the gathering of men in the Finch's front yard the day before. Needless to say, Atticus was sadly mistaken, because the KKK was quite active in Alabama at the time of the story, and the group is still around in force during the 21st century. As for Mr. Cunningham, he was simply the leader of the group of men who had decided to lynch Tom. No KKK member, he was just a man who had had too much to drink and who had murder on his mind--at least until Scout managed to change his mind.
Thanks you so much.