I’m going to go against the pessimism here and say that this is the US not Europe or the UK and the First Amendment has teeth. There’s ample Supreme Court precedent that anonymous speech is a protected right (Talley vs CA, Macintyre vs Ohio, etc) so I’d expect efforts like this to flounder in the courts if push came to shove.

What the US Supreme Court decides is much less relevant than it used to be because the US executive can and will simply ignore the decision. If anyone in the US administration breaks the law, they can be pardoned by the US president,if the president breaks the law, he's immune against prosecution based on a previous decision of the Supreme Court, and no court can enforce anything if the executive doesn't comply with the court order.