This is more akin to the chief of police making an announcement that a building is safe to enter, knowing there is a gunmen ready to shoot people as they enter. I don't know the exact definitions here, IANAL, but I'm pretty sure my scenario would be legally actionable.
Similarly, freedom of speech is limited at least in one way by "imminent lawless action."
I'm not sure about the issue you mention regarding CBP, but I believe there are certain differences between how the law treats citizens and non-citizens. The Exclusionary Rule, disallowing evidence illegally obtained, doesn't apply to civil matters, including immigration, apparently (https://www.halt.org/things-cops-do-that-are-illegal-what-a-...).
Law enforcement can omit or lie about your Miranda rights, but they can't use evidence obtained from what you say without reading your rights. So if a cop saw you shoot someone, and you're still holding the gun, they don't really need to read you your rights; they don't really need anything from you at that point.
Is it not illegal to get someone to sign a contract based on a known falsehood? I.e. if I offer to sell you my house and tell you that it has no issues, but it turns out termites have eaten half of it and I knew that, is the contract not invalid? I think your solution only works because it's common sense that that is not a statement of fact.
I understand why they do it. But I am 100% for anything to increase police accountability. They should be held to a higher standard and obviously that is not the case today (at least in the United States).