Yep, the accusation is all we need to render a verdict and sentence. That's our great American legal tradition!
What's going on is a concerted effort to eradicate christian beliefs from the public sphere, IMO. This case is a textbook example of demanding special (not equal) rights under the law.
Then tell me, what religious principle convicts her of engaging in sin if she processes this document.
In the article it states she believes the Bible condemns homosexuality as sin, and therefore her personally signing such a document would be a sin for her, because she would be supporting the sin of others. Whether you agree with that position or not, that is her viewpoint according to the article.
Then just by being part of The office of the Clerk, she is also supporting it. Even you acknowledge that since, as you said,
" The clerk herself and the clerk’s staff are (or should be) the same legal entity, as the position is not considered to be an individual, but an office.
If "The Office Of The Clerk" is the issuing agent, and what they issue is supportive of homosexual marriage, then anyone who who is part of that office also supports it.
If that doesn't quite sound right to you, then neither does the notion that a signature, or stamp, or whatever it is that they do to process this paperwork, is tantamount to support either.
Take your pick. It's one or the other.