Your statement that his signature confirms the state’s approval of same-sex marriage is significant. He is representing the state and not himself. I doubt that anyone who has been married in your county cares that he is the one whose signature appears on his or her document of marriage. I doubt that they even notice the name, let alone know whom it signifies.
He didn’t just happen to get that job; it is the way he makes his living.
Personally, I would not be thrilled about having my name on the marriage documents of gays—nor anyone else who is not validly married, including the many divorcees on whose documents his signature appears. I think you need to let this one go.