Nah. Klan never acted outside of their safe zone. Hoover never really left DC less... well he never really left once he was in charge.
So if we have a FBI-Klan war how does this effect the south overall? We had racial based reforms done via legal means, the FBI active in combating Klan terrorism in many ways was targeting the most staunch of Southern states secret power structures. More over this means that almost anyone from Marshall to MLK are now open targets.
Honestly, if the Klan managed to survive COINTELPRO and a spate of assassinations in good enough shape to keep fighting the FBI, it'd also mean the National Guard would probably be called up, and used as extra muscle for FBI operations, and providing law enforcement in certain areas. It might even get serious enough (with enough problems involving unreliable state and local authorities) that Regular Army units would be deployed under the authority granted to the president under the Force Acts, as had been done in Little Rock IOTL.
That of course, would not go well, but in the 50's there should still be a decent amount of institutional memory left over from the Banana Wars in the USMC, and someone should remember the Small Wars Manual, so within a couple years, the Klan is shattered and impotent, and power structures in the deep south are once again fixed.
Considering all of that, I could actually see Law Enforcement in the affected states be completely replaced by a Federal/National Police Agency or Gendarmes (like the RCMP, either way) for a good decade or so, until new police departments can be established and staffed with reliable (and often black) officers. The Justice Department would probably also start vetting candidates for Sheriff and Police Commissioner offices in the affected region as well.
