Remember, both MLK Jr and his father were registered Republicans for most of their lives. They belonged to the party that gave them freedom and the right to vote.
One could not register by party in GA at that time, and there were no Republican primaries there until at least 1968 or later. The Kings backed JFK and LBJ. I don’t think they were for Nixon for governor of CA either. Were they for Eisenhower in 1956; I don’t know. I don’t think people can register by party in GA even now; their party allegiance is determined by which party primary they vote in; the same as in TX.