Such a law should also clearly provide that any candidate who fails to provide absolute and unambiguous proof of NBC shall not be listed on the ballot for either President or VP and that even if elected (through printing error or write-in votes) shall not receive any electors;
and that any candidate which provides false, forged, or otherwise tampered with documents, false testimony, or any other false evidence shall be guilty of a felony offense, penalty of which will be no less than 20 years at hard labor, and prima fascia evidence of treason;
and that electors representing the State shall not cast any vote for such disqualified candidate;
and elector casting such vote shall be guilty of a felony offense, penalty of which will be no less than 20 years at hard labor, and prima fascia evidence of treason.
And considering that the states control their ballots and can set their own criteria for who can be on it then there is no reason why such a law wouldn't be upheld by the courts. And it would take only one state to pass a law for it to have its desired effect - no candidate from a major party can refuse to run in any state without eyebrows being raised.