Good question.Then why is (and was) the Klan so eager to closely associate themselves with it?
I would say that the flag, the Klan and black people have interacted on the issue over time. The KKK does not like interacting with black people, black folks don't like the flag, so the more that black folk don't like the flag, the more the KKK carried it. This was especially true for the 1940s on.
Here is a photo of the Klan parading in Washington DC in 1925. Note the flag.