[Always Right]

Would it be permissible to establish a U. S. Department of Religious Advice whose mission is to study and evaluate religion and issue non mandatory religious advice?

You are getting beyond ridiculous. Allowing religious speech is Constitutional. Trying to establish religious doctrine is not.

[FredFlash]

Even if the Department of Religious Advice did attempted to establish religious doctrine, but did not use coercion, it would be permissible under your no coercion test - right?