Every time some case like this comes along, the legal positivism bandwagon clatters through town. Goodthinkers argue that clerks and other public officials must - they absolutely must - uphold the laws exactly as they are written no matter their "personal feelings." The personal feelings line is actually meant to discourage us from doing any hard thinking on whether there is a conflict between the positive law and the natural law.
Ironically, there is nothing in the United States Constitution or any statute anywhere in the English speaking world that states we must embrace legal positivism. It's actually a fairly recent innovation in Anglo-American judicial philosophy, and not one that is without critics besides religious wacko extremists like me.
The natural law is the necessary basis of positive law:
1959 The natural law, the Creator's very good work, provides the solid foundation on which man can build the structure of moral rules to guide his choices. It also provides the indispensable moral foundation for building the human community. Finally, it provides the necessary basis for the civil law with which it is connected, whether by a reflection that draws conclusions from its principles, or by additions of a positive and juridical nature.Other things being equal, we have a moral obligation to obey civil laws even if our rulers are wicked. We may elect the cryogenically frozen brain of Hitler president, but we still have to drive on the right side of the road. However, that obligation ceases and we gain a moral obligation to disobey any civil law that purports to contradict the natural law. "I was just following orders," is not a valid excuse to do evil.
The choice that is presented to Christian public officials - either enforce the law or resign - is a false dilemma that begs the question. There is no problem if you do not presuppose legal positivism. The left-liberal option of jumping out of the positivist frying pan and into the emanating post-modern penumbral fire is also false. Both represent attempts to push the natural law outside of the Overton Window and replace it with the Nietzschean will to power.