Except that you have to *prove* I'm lying. It can't just be because Reasons. You'd need to catch me in illegal possession of the weapon in question.

Sure, but that's not really all that hard in the situation you've described. You own a gun. There's almost certainly readily available documentary evidence that you owned a gun in the past, and it's a simple matter to provide in the law that prior owners of guns have to provide a statement describing the disposition of that weapon. The police ask you what happened to your gun, and you provide a transparently false story - one that will probably fall apart even with a few follow-up questions, and certainly with even a minor amount of effort to confirm it.

Super-easy at that point to get a warrant to search your house and find the weapon that you've been illegally keeping. It's not hard to catch criminals if their sole plan for getting away with their crime is to tell the police that they didn't do it.

To do what you want, you're going to need to lock the border down.

Why? Guns going to be especially poor candidates for smuggling. They're very large and heavy relative to their value - especially guns like an AR-15 or other semi-automatic rifles. Current prices for guns are roughly $10 per ounce. That's an order of magnitude lower than even marijuana (which is well over $100 per ounce even while being de facto legal, but used to be smuggled in quantity); actual illicit drugs like heroin or fentanyl are about $6,000 per ounce. It's theoretically possible that there might be a market for $10,000 pistols, but it's unlikely to be particularly large.