OMG, it's fraud! It has nothing to do with the constitution other than the fact it is irresponsible and should be treated as a criminal act. Lying is subject to contempt in a court of law. Raise your right hand.
So is wearing a padded bra or pretending you are a surfer when you are not (see The Beach Boys).
Fraud is not generally a crime unless you do it for the purpose of stealing someone's money. If you do it so that you can impress chicks, it is constitutionally protected speech.
Lying is speech as much as telling the truth. If this guy had committed Criminal Fraud by pretending to be a war hero for the purpose of getting someone to give him money, then he could and should be prosecuted for criminal fraud. But there are already laws on the books about that.
It’s fraud if in doing so, the liar seeks material gain at the expense of the lie-ee. For example, what if an alien runs for president under the pretense of being a U.S. citizen?