Do a little research, and you'll see the FBI was the only group investigating KKK related in some places from the 1920's though the 60's. The KKK is nothing like it was. And don't forget the context of the cold war. Communism was an existential threat to the status quo, in a way the KKK never was.
The KKK was targeted by the FBI but not on the same scale. The Panther’s Fred Hampton was murdered by the FBI in his bed. AIM’s Leonard Peltier remains in prison to this day. They classified all the mentioned groups as “hate groups”, Including the klan, but also ridiculously put the non-violent, non-nationalist SCLC and SNCC on the list of “black nationalist hate groups”.