False claim? How about we call them lies? Trump is a liar. He lies.
A false claim implies it could be made with good intentions, simply out of ignorance. No. Trump lies. He lies maliciously.
How many lie-filled years need to pass before media call it what it is?
I think it should definitely be pointed out, constantly, how much of a hypocrite and liar Trump is, just to get it (and keep it) on the record. Ignoring it would only normalize it more. And maybe some fence sitters didn't know about it and now they do. Which is a good thing.
At the same time, however, I also believe it's absolutely pointless, if your goal is to convince any of his followers. They already know all of this and they don't care. They don't love him despite the lies and hypocrisy. They love him because of it. And even if they don't like it per se, they still won't turn against him, because they're accepting way worse and nothing has been a deal breaker so far.
Edit: I have often wondered what Trump would have to do for his followers to abandon him. Is there anything they wouldn't accept?