Nonsense. The Army (last I heard) is part of the Department of Defense. The DoD is part of the Executive Department, which is under the President. So is the Department of Justice, which is over the BATFE, and the FBI which operates the NICS check system.
All of those take orders from the President, and are subject to Executive Orders. I don't know if an EO could be used to stop BATFE from issuing FFLs or to stop the FBI from doing NICS checks, but trying to do either would create a real kerfluffle.
So the idea that BATFE, or the FBI, or the Army, or CMP, could or would override an EO is silly. It would be like the office boy telling the CEO what to do.