In my opinion don’t need an amendment. Congress right to raise an army gives Congress the right to criminalize stolen valor. If they pass a law that is phrased in that manner the courts will uphold it.
We have a long history of jurisprudence on the 1st Amendment, so the types of things that are not “protected speech” are pretty well defined (with the notable exception of “obscenity”, which is only loosely defined).
So yes, you’re going to need to pass an amendment if you want to add an entirely new class of speech to the “unprotected” category.