Answer by Kris Rosvold:
She has a job which she is paid to do.
Here's the Job: Accept notice on behalf of the State, and make a note in the official records, that a couple is getting married and that they are legally entitled to do so.
There is no ethical "dilemma".
If she insists that there is one (such as I would have working for ALEC) she can either do the Job in exchange for her salary, or find a different job which doesn't offend her morals.
She (under the First Amendment) doesn't get to try to enforce her morals through her government job.
As with so many zealots she seems to conveniently "forget" that while the protections of the First Amendment don't limit her beliefs ^in her private life^ they DO limit ^her application of her beliefs to her Job as a member of the government.^
Her refusal to sign could easily be seen as a defacto attempt at creation of law enforcing her religious beliefs upon others.