The answer is “it depends.”
Yesterday, there was a post about a suit from a Muslim air stewardess who objected to serving alcohol on the plane, although it is a clear duty of the job. The question was, “would arrangements be made that would enable her to perform her job without that doing that part of it?”
If the answer is yes for the Muslim stewardess, it should be yes for the Christian Clerk or Justice of the Peace or cake baker.
The two issues are completely different.
The stewardess knew about the requirement coming in. Customers are responsible for their consumption, and she does not have to authorize the consumption under her signature.
The clerk had a federal interpretation of law changed, without updating the KY laws she is beholden to, that requires her to personally authorize the marriage in her own name.
There are no parallels here.