White supremacists in the South aren't going to blow up into a full paramilitary insurrection so long as they've got the local power on their side. Perhaps the Germans before and during the Second World War send more feelers into far-right groups and get enough of them on a program of 'dividing the world for the Aryan man', with the Nazis having Europe and themselves turning the Americas into a protestant white-dominated slave empire, with the Jews purged and the blacks and Latinos put in their 'place'. With a Republican president and Congress with at least a soft civil rights agenda and Germany being a bit less overtly antagonistic to the US, and supporting their guys in the South with money and weapons, you could have them exploding into partisan resistance against the American war effort. This could be enough to produce the political will to initiate a second Reconstruction that would lead to a continuing underground network of white supremacist paramilitaries.
Alabama would want to take every measure possible to keep things from getting bad enough that Washington would feel justified in and face public pressure to get involved. If anything would get the country to gather the political will to actually crack down on Jim Crow, it's white supremacists running amuck with absolute disreguard for even the venere of rule of law. Showing absolutely no willingness to bend is liable to result in being broken by the imposition of marshal law and the end of white supremacy,while cracking down on the worst excesses allows the main body of the policy to survive.
Maybe king is killed earlier and Malcom X survives leading to a far more by militant movement which lead to a stronger klan in return. The KKK would attack blacks in the south leading to X sending armed men to fight back against them. I could see it turning in to tic for tac like Northern Ireland. This is not even mentioning the soviets or China, Cuba who would love to “send aids to black freedom fighters” if it get really bad there could even be used of chemical weapons on both sides.