Mrs. Davis is performing the age-old act of civil disobedience. It's amusing to see the angry mob now calling for the judge to put her in contempt, jail her, fine her, anything to punish her for refusing to put her name on the document.
Honestly, there has to be a compromise. If she is that uncomfortable, they could agree to sign it with the words "State of Kentucky" or "Rowan County, KY" or something that takes away the very real concern of having her name associated with this. But as a clerk, she has to perform the duties required of her—so eventually just flat-out obstructing the law is going to have to result in punishment or her resignation eventually.
The article duly notes that Davis, as an elected official, can only be removed by impeachment, and because the legislature's not in session, she can't be removed. It really is an ingenious way to approach it. If Kentucky truly wanted to nullify the court, they could simply turn a blind eye to Davis and others who refuse to do so... and there'd be very little they could do about it.
