This woman is an elected official. She doesn’t answer to anyone except the voters. If she doesn’t want to sign a license, I cannot see were she would be in jeopardy except in her next election.
If this was just another some elected official who didn’t feel like doing something, say a sheriff who wouldn’t respond to emergency calls, would they be in any sort of trouble from a higher authority? No. Just the voters.
>>> This woman is an elected official. She doesn’t answer to anyone except the voters.
And courts. Being elected doesn’t put her up above the law. I expect there are precedents from the old civil rights era with officials refusing to grant marriages papers in mixed race cases.
Assuming state law doesn’t grant this office the option to pick and choose which citizens they may serve, it’s open and shut. She will lose.