I gotta doubt that that's feasible:
2017 R-Pod RP-176 is a very small travel trailer weighing dry in at 3800lbs/1720kgs, it's not the trailer in the photo but... It looks about the same size, single axle, short as shit, tear drop shape. And that's the dry weight, once you add in gear That's a lot more weight. Typically most people expect to add in about another 1500 lbs/680kg from the dry weight. So that's a total of about 2400kgs, which is past what that car should tow.
In fact I'm not really seeing any trailers that aren't pop-ups that would qualify at the weight you're saying, even without gear.
That's not to say it doesn't exist, hell maybe my Google searches aren't showing me shit from across the pond that would work, but metal is heavy.
So what? Add another question to the background check "are you a Nazi?" because the "are you a terrorist?" question should already take care of that one, so it seems to be working really well.
What about how I can just gift a gun to a family member? Or maybe how no one has followed up on me since I've purchased a gun, maybe I'm a crackpot now. People change.
Responsible gun ownership is impossible to enforce, sure my shit is locked up in a safe, but no one knows that and it isn't required. No one comes around to check that it is safely secured, no one comes around to check that I haven't fallen down a QAnon conspiracy rabbit hole, no one comes around to check that I'm mentally sound and don't have any anger problems.
Ammo sales aren't tracked either, so sure I have a reasonable amount of ammo but I could have enough to cause serious problems as no one would know.
So really this comes down to two solutions right? Take away guns, or spend a fuck ton of money to regulate sales, registration, send therapists to people's homes to evaluate their mental health periodically, track ammo sales, install biometrics on guns so that only the owner can fire it and as evidence of when the gun goes off who must have shot, install trackers and sensors on gun safes to feed information back to the government to know how often a gun is safely locked away and when it's not where it is... Etc honestly.
And again, I'm a gun owner.