This letter is from a leading defamation law firm but it does not actually threaten legal action, it just asks the Times to "do right by its readers" (maybe that's after the first four pages that are shown in this tweet). If they thought they had a good case, they would make that very clear. The fact that they spend a lot of time complaining about things that the Times didn't actually report and complaining that he was treated differently than other people tells me that this is all for PR.

written by an independent author /s

"Hoax" is a popular word coming from this White House