Actually, it was the Kennedy Administration that had King investigated- not "conservatives" who were out of power for many decades at that point in time.
Libs preferred the rhetoric of the civil rights Left Wing led by Stokely Carmichael and H. Rap Brown. MLK was looked upon as an Uncle Tom by his liberal contemporaries as he both dressed and acted white. Look at photos of the fellow, never wore hit pants "sagging", never called people "mutha fuckas" , never smoked newports or drank malt liquor in public.