I can't address how the law sees it but from a principled standpoint, I don't see how a person acting as an agent of the state and employed to do a ministerial task is violating their religious beliefs by doing so. They're not asked to bless, approve, sanction, ANY marriage. Their sole task as a state/county employee issuing licenses is checking off a few boxes, and if the couple meets legal requirements, issuing a license. And given that it's certainly not "religious animus" to expect a person employed by the state to simply do their job instead of butting their heads into my choice of spouse or my religious beliefs.Why do you feel this need to force someone to do something against their religious beliefs when an acceptable alternative can be worked out? The only plausible reason seems to be anti-religious animus.
And related to the point I made earlier, if a clerk is actually motivated to only issue licenses in accordance with his or her deeply held religious beliefs, we should expect them to quiz all straight couples coming before him about their religious beliefs about marriage, and promptly refusing them a license if her own church would refuse to conduct the ceremony, as best she can tell in a 60 second or so interview. We all know that isn't going to happen because straight couples being subjected to arbitrary religious tests that vary from clerk to clerk wouldn't be tolerated in that office or in that town or county. We could measure their careers in that job in hours or days, not weeks.