She’ll lose. It doesn’t matter is she objects to the marriage. All that matters is that the government allows. A license does not signify your personal approval - just that the requirements of the government have been met. Her job is solely to see that government rules are followed.
If I help someone on their tax return, and they are homosexuals, they will still be eligible for married filing jointly. My opinion of their morality is irrelevant.
I suspect this case is being taken where others (baking cakes) have not because it will be easy to slap down the religious objections.
And who you gonna serve? Bob Dylan said it well,
"It may be the devil
It may be the Lord,
Buy you gotta serve somebody."