In my own life I have also had this struggle, especially around Christmas time, when there is extreme pressure to conform. I can remember a time when suburban Jews had "Hanukkah bushes" just to avoid being harassed by Christian neighbors. Ditto for putting up "holiday lights." If you were the only family on the block that did not, you heard about it. It was simply assumed that everyone was Christian.
Now the shoe is on the other foot. Puritan Christians are no longer in control. And suddenly people like Kim Davis are finding out what it is like to be asked to do something you do not believe in. Only with a difference. Because Judaism has a principal called, in Hebrew, dina malchut dina, "The law of the land is the law." Jews do not insist the whole world live according to our beliefs. When it comes to secular issues, we obey the laws of the country we live in. In the case of Kim Davis, as a public official, Judaism would say that she is required to issue those licenses.
I have tried to think of a similar issue for me as rabbi and came up with this: Jewish law forbids intermarriage between a Jew and a non-Jew. Many synagogues will not accept such families as members. That is their right as religious institutions. But what would I do if such a couple showed up at the courthouse asking me to issue a license, and I knew for a fact it would be a "forbidden" intermarriage? I would issue the license. I would not perform the ceremony. I might not go to the wedding. But in terms of the secular law, I would be required to fulfill my duty. This would not be a breach of my own faith, since I would be doing it as a public official, not as a rabbi.
Christians need to learn how to make this distinction between public and private. They need to accept the fact that are not the majority anymore, and they no longer run the whole show. Kim Davis could best demonstrate her faith by resigning her post and seeking another job.