Is religious freedom vs. equality under the law as in the Kim Davis situation an issue whereby we must obey God rather than men, or is the believer subject to obey the law when elected to uphold the law?
The situation you are referencing is a good reminder of how challenging it can be when Christians live in two kingdoms: the church and the state. While God’s word guides life in the church, natural law and human reason are to guide life in the state. Sometimes the state enacts laws that are contrary to natural law and human reason. What then? Is the state, the government, still God’s servant (Romans 13)? Yes, but not a very faithful servant. God requires our respect of and obedience toward government officials not necessarily because they deserve respect and obedience by their actions but because governing authorities owe their existence to God, serving as his representatives.
Christians who want to serve in the state can do so—even when laws run contrary to God’s word—by recognizing that the state operates on different principles than the church. Moses recognized that. As the leader of the church, he penned under inspiration of the Holy Spirit the establishment of marriage as a lifelong union between a man and a woman (Genesis 2:20-24). As the leader of the state, he wrote certificates of divorce (Matthew 19:3-9) that served to preserve order among Israelite society. A Christian judge might find himself in a similar situation: being in full agreement with the Bible’s teachings of marriage and divorce (perhaps even taking part in a congregational vote on excommunication related to an unscriptural divorce), yet granting an unscriptural divorce in his court. Such actions on his part are not hypocritical; they reflect his involvement as a member of the two kingdoms of church and state that operate with different guiding principles.
In the situation you referenced, we are not talking about a judge issuing a final decree of divorce for unscriptural reasons but a county clerk refusing to issue marriage certificates to people whose union is not biblical. While same-sex marriage is ungodly, it is legal in our country. Christians who find themselves in a situation where they are legally bound to issue marriage certificates to people of the same sex can take the approach of Moses or the Christian judge mentioned above, or they may take a different approach because of reasons of conscience.
Here is where approaches will differ among Christians as they explore various options. One of the options available for the person cited in your question is to seek a different vocation where conscience would not be disturbed. Even though she was elected to her position, she is not forced to continue in her office against her will. It would be very understandable if she resigned for reasons of conscience. It would also be understandable if she followed a different course of action for reasons of conscience and suffered whatever consequences might follow.
The bottom line is that there is no easy, one-size-fits-all answer to the question you posed. Christians will prayerfully seek to determine how they can best serve and honor God as citizens of both kingdoms, recognizing the principles that guide each kingdom and applying them as faithfully as they can to their situations.