The fact is that under federal law, a private sale does not require any background check, and does not need to go through a dealer. I do not know why you think you can get away with such a blatant lie, but I am willing to bet you will now change the claim to the gun show loophole is somehow good.
https://www.findlaw.com/consumer/consumer-transactions/private-gun-sale-laws-by-state.html