The President of the United States is an agent of Congress as "chief executive" over the U.S. government agencies and departments and being that the president of the U.S. Senate is his underlying he is part of Congress. That is, a president presides over Congressional sessions. Who has the veto power? Whose underlying is president of the the U.S. Senate? As a commander in chief he is a commissioned officer--a soldier. It would follow therefore that President of the United States is also prohibited from making laws respecting establishment of religion.
I meant to add that you do make a very good point in that one would be wise to be mindful how one approaches a given matter or venue.