the only thing outrageous about this is that they think MLK was a Republican.
MLK Jr. never did express a personal preference one way on the other. Yet I think it would not be unfair to say that up till around 1960, his inclinations were towards the Republicans. However the maneuvers both parties took on civil rights meant that he likely was Democratic in sympathy by the end of his life. Hence it would make as much sense for Republicans to claim MLK as it would for the Democrats to claim Reagan because of his political preferences early in life.
By most of his writings, he was a democratic socialist.
And in the South of the 1950s, which party would come closer to that? The Republicans or the Dixiecrats? The Republicans hadn't yet put African-Americans completely under the bus in their quest to pick up the votes of southern whites.