Help Me Out Here: Am I Missing Something?

I came across the meme on the left on a university friend of mine's Facebook newsfeed and I'm having a really hard time comprehending this one. As I understand (and, please, please, please, tell me if I'm interpreting this wrong), I should support the Muslim county clerk's decision to deny a customer a marriage license on the basis that the customer is not wearing a hijab and that, in not supporting her - which, in this case, I don't - means that I'm supporting Christian privilege.

Firstly, is it not the same religious right of the woman customer to choose whether or not she wears a hijab? Secondly, is it not the actual job of the Muslim clerk to provide a service (be it supplying marriage licenses or something else) to the town she serves?

I do support the autonomy of the clerk to actively find someone else to help the customer. But if no one else can be found, the clerk's refusal, while religiously understandable, seems unethical given her job and the freedoms that are supported by the American Constitution, under which she serves, as a working person in America.

Please, please, please let me know where I'm going wrong. I feel like there's something simple that I'm missing, but I can't help feeling that this meme (as many obviously are being that they're, you know, memes) is grossly oversimplifying this issue and eagerly pointing the privilege finger. And I'll be honest, I point the privilege finger a lot because it's worthwhile and it is a privilege to be taught about privilege (or have your privilege checked) as opposed to experiencing a lack of it. So, I beg of you, tell me where I'm going astray in my logic.