Cerb - you seem to be missing the difference between the separation of church function governing a state, and the state being under the obligation to not make rules guided by religious doctrine. Those are two very different things, but you're basing your argument on them being one in the same.

There is nothing in the theory of separation of church and state that says the state can't mandate a person be allowed to wear a haqib in a mugshot - there IS something in the theory of separation of church and state that requires a person to wear a haqib in a mugshot.