I like you hoosier, you are extremely different than me, but you have always posted within your "I am a Christian, here is what I believe" persona. Putting aside the clerk's probable hypocrisy (divorced persons), I'm with you, I just don't know what she is basing the objection on.
Even if she believes it is a sin, is issuing a State (Non-Religious) license, "participating" in the sin? Is using a stamp to mark a deed, participation in cohabitation? This is simply statistical, clerical work, imo.
Edit to discuss your very first comment: I don't believe homosexual marriage is legitimate in the eyes of God...
Of course I don't think government should interfere with your belief of this, but does a recognition that someone is married NOT in the eyes of God, but in the eyes of Kentucky interfere with that belief? It seems to me that you, the clerk, anyone could do both of those things simultaneously.