Can someone help me with a possible false vet status claim?

If the guy was just some smackhead working as a Walmart greeter I wouldn't care.
The man is running for office and part of his platform is being a Vet - so I have a very specific interest in finding out the truth.

PO1 Nathan Watts - Easiest way is to ask him to provide a copy of his DD-214. If he's reluctant or refuses, you'll have your answer.

In your place I'd tread carefully here.

There are a lot of folks who have some psychological need to embellish or outright lie about military service, and I see nothing wrong with pointing them out; especially if they're trading on their "veteran" status to win elected office. That said, too often the Guardians of Stolen Valor--self-appointed of course--get it wrong, and when they get it wrong, they do so publically and a legitimate Sm or vet has to deal with the consequences. I'm glad you are looking for something tangible to cite; whereas too many just make accusations based on mere feelings.

If this guy is lying, I hope you find evidence that backs your inclinations, and you rightly call him out. If he's legitimate, then it is what it is, and no accusations need be made.

However it shakes out, good luck.