Something I've played with is to have the KGB promote civil rights groups, ideally even outright black nationalists. At first, they're blacks getting ideas above their station in life and causing trouble. Then, once their ideology comes out, they're REDS trying to overthrow the lawful order of the United States. And have the KGB back groups like the KKK too, to get a nice little race war going, but do it more subtly. The trick is to keep it a bleeding sore, make sure any time there's a viable peace process going there's a major atrocity committed.
Such an ongoing terror campaign - vaguely similar to Northern Ireland - could be used to justify considerable McCarthyist paranoia. It also has the useful fringe 'benefit' for a dystopia of associating civil rights with communism. Expect to see Jim Crow, highway checkpoints around 'high-risk areas', civil rights advocates drawing the full wrath directed at anti-communists, the whole lot.
Oh, and potentially the US Army sent in to quell insurrection, weakening Europe and Asia. Though it would probably mean counterinsurgency doctrine is refined, which might help in Vietnam. If, of course, the continued Red Scare doesn't result in the US just bombing the place flat.
All of this depends on the KGB being more effective at operating inside the US, though - or the FBI less effective at catching them.
The two problems with all this?
First, you already HAD J. Edgar Hoover believing that the whole civil rights movement was being directed straight from Moscow, as his beliefs taught him that black people were just too simple minded

to do anything so complex as organizing things like sit-ins and freedom marches.
The one or two actual communists who were active in the civil rights movement were used by Edna as "proof", since they were obviously Moscow's direct liaisons. They, and some of the traitors to their race white people among them.


The fact that one of the long time leaders of the CPUSA was black was used by Edna too. But the civil rights movement came too late for Edna in terms of getting a friendly hearing from whoever was in the White House.
The last POTUS who would have given him a good listening was Woodrow Wilson, and by the time of his stroke even the NAACP had only been around for some five years, and Marcus Garvey was still several years off, never mind W.E.B. DuBois, never mind MLK jr.
But the real stumbling block to all this was the second problem: Internal politics within the KGB itself and the USSR political structure in general. For them, having KGB involvement in the US civil rights movements would be a Red Badge of Honor, not any embarrassment at all. The nations of Africa, Asia, and South America would cheer them on, and Europe would be anywhere from that to a muted silence.
But if the KGB were caught redhanded supporting the KKK?

Worldwide condemnation singing "hypocrites!" over how the USSR had spent decades demonizing the USA for its murderous KKK activities.
If the Politburo actually approved this (and I can't possibly see how they ever could) you could see major overturns in power in the Politburo and the Central Committee.
If the KGB (or more likely some rogue faction within its secret corridors of power) goes it alone on this scheme, you could see Krushchev or Brezhnev sending the KGB Chairman to a Siberian dacha, the Chiefs of the individual Directorates involved sent to count trees in Siberia, the middle level commanders to dig in the salt mines, and the people directly involved in the operation lined up against a wall and shot!
In the eyes of the Soviets, the KKK was seen as the Ultimate Evil of the Dark Forces ruling America. And Soviet propaganda was quite right in this case. The best political cartoon I ever saw in my life was Soviet. Most of their political cartoons were very primitive, trying to say a dozen different things in just one caption. But i saw one with a white southerner dressed in the all white ice cream suits popular in the South at that time. He is sitting back in a lounge chair telling a reporter that he condemned South African Apartheid as being racist.
Up in the high left hand corner of the cartoon was the bare feet of a lynched black man.
Devastatingly effective.