Dr. King wanted people to be judged not by the color of their skin, but by the content of their character. Affirmative Action seeks to judge people by the color of their skin and disregard the content of their character. Affirmative Action is hardly a construct of conservatism.
Yet, he pushed for affirmative action. That must mean that affirmative action is not how you characterized it. I hope, by the way, that you do not defend racial profiling.
King called for judging people by the content of their character, and for affirmative action, but not at the same time. The two notions are incompatible. Some time after the 1963 Poor People's March on Washington, King betrayed the color-blind ideal he had championed, and which made him so famous and admired.
At least that's the charitable interpretation. A less charitable interpretation would be that he never believed in character, but paid lip-service to it as a weigh station, on the way to color-based quotas. In any event, you can support character or affirmative action, but not both.