I'm curious as to how you think this would work. Would it only apply to the President? What about the Vice President? What about cabinet members or other unelected officials, such as government spokespeople. What about members of Congress? Would it apply to candidates for election if they aren't already elected officials? Could someone challenging an incumbent lie without consequence, but the elected incumbent would face potential criminal liability every time they open their mouths? Do you really think it would be a good idea to let Bill Barr's Justice Department decide whether or not to launch a criminal investigation into elected officials? Every time someone accused an elected official of something and the official denied the accusation, it would set the stage for a criminal investigation.

So, no, it shouldn't. We have the First Amendment. We have the Speech and Debate clause. We don't need to re-write the Constitution.