I think I have to disagree a bit with how this question is framed. King describes the difference between just and unjust laws in the thirteenth and fourteenth paragraphs of his letter, but nowhere does he discuss some distinction between moral laws and natural laws, and in fact, he seems to regard them as much the same.
King's distinction between just and unjust laws is important for his purposes because this distinction is the justification for his civil obedience. A just law, according to King, is one that "squares with the moral law or the law of God" (King para. 13). An unjust law "is out of harmony with the moral law" (para. 13), and "a human law that is not rooted in eternal law and natural law" (para.13). A law that is degrading to the human spirit is unjust. A law that a majority forces on a minority without being subject to the law itself is unjust. A law that has been enacted without any representation upon the people it is imposed upon is unjust. Implicit in his argument is also that a law that separates people from one another is an unjust law.
King's campaign of peaceful civil disobedience rests upon his argument about just and unjust laws. Whatever distinctions there may be between moral and natural laws, King does not find it necessary to expound upon them.