It’s not up to a civil servant to determine which laws she will follow, repulsive as the associated action’s request may be. I believe the saying is “I don’t pay you to think.” If conscience dictates she can’t perform a duty, she should resign.
This situation is quite different than a priest refusing to give the sacrament or a doctor in a Catholic or other religious hospital refusing to commit an abortion. Throughout history individuals have payed far greater prices for maintaining their convictions.
A solution might, if this were an elected position, for her to resign and then run again with her beliefs in full view. If she were to be re-elected with support of the population, she would be able to claim she represents the will of the voters.
The established rule is that ALL marriage licenses are signed by appointment. The pair refused to wait for an appointment.
Did this little fact escape your notice?