It's not actually clear that Musk was involved in this at all.
By all accounts, he has not really been present at the day-to-day operations of SpaceX in years. And the open letter was targeting the executive team, not Musk.
While we don't know if Musk was involved in this legal approach, it absolutely reeks of Musk. What corporation, other than an Elon corporation, would make a labor rights case about the constitutionality of a century-old labor rights department?
“Elon’s behavior in the public sphere is a frequent source of distraction and embarrassment for us, particularly in recent weeks,” the employees wrote. The letter, which welcomed other employees’ signatures, called for SpaceX to distance itself from Musk’s controversial online comments.