4th "The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized."
They require either probable cause or a warrant to search you, or examine your "papers", thus it has always been, and is supported by enumerable SCOTUS cases and lower court rulings. That you can site a few outlier rulings from idiot judges (some judges ARE idiots) and pretend that negates the 4th Amendment is lunacy, but our system of legal precedent is pretty loony standing all by itself. Simply doing what is legal isn't justification for a search, no matter how much of a mindless drone you are. The potential to commit a crime, or to not be in compliance with some authoritarian law never has been justification for a search (if it was then the whole concept of "probable cause" is out the window), and the day it is there will be a revolution, because everyone will be a "criminal" but those with a badge (some already see the world this way). This is a very basic tenet of our rights as free citizens, and without it we are not citizens but rather subjects. Now, practically, the police can do whatever they want, and prosecutors can prosecute whatever they want. There are stories of this everyday, and sometimes they even win their lunatic cases when they get a similarly lunatic judge. Ultimately you have to be willing to take the time and spend the money to fight in court, and they are banking that you won't. This is how our liberty gets sapped away one unlawful search at a time. We are predisposed to it, as we are more likely to suffer evils while evil is sufferable that to right our selves. You are kidding yourself if you think the apatite of the American people for suffering tyranny is bottomless.
No one wants criminals to get away with crimes, but trading that for assuming everyone is a criminal is way worse than letting a few get away, and the moment it isn't this is no longer America. Nothing disgusts me more than law and order types who claim patriotism with no respect for what makes patriotism worthwhile in the first place, LIBERTY. Without that you're just a henchman working for the powerful to keep them powerful. There is a reason we swear loyalty oaths not to a government or a set of politicians, but to ideals enshrined in a document, and you don't understand the oath if you believe that orders supersede those ideals you're in the wrong business. No one gives up the right to think and be a moral person because they are under the authority of someone else, and if you have been so convinced you make yourself an enemy of everything this country was founded upon and everything we believe. Without your ignorance and complicity our system keeps (or at least used to keep) tyranny in check, with it we are in deep trouble.
It requires belief in the system, and belief in human liberty and human dignity. If you don't believe in those things you have no business claiming to serve them or following the orders of people who don't.
From the ATF site:
"The approved application received from ATF serves as evidence of registration of the NFA firearm in the owner’s name. This document must be kept available for inspection by ATF officers. It is suggested that a photocopy of the approved application be carried by the owner when the weapon is being transported" The stamp is (was) affixed to the document.
Another LEO can probably arrest based on the suspicion of an illegal firearm if you can't produce the same, but only the BATFE can force you to produce it as a condition of it's issuance.
Like all these draconian laws there is necessarily a huge gray area of questionable legality in which nefarious forces can play to deprive you of your pursuit of happiness. In that respect they go against the spirit of the Constitution and the Declaration of Independence if not it's letter. If you value liberty and the free pursuit of happiness these are precisely the kinds of laws we should all be working to banish to the dust bin of history.