In compelling Senate testimony, Yale law professor Akhil Amar explained the constitutional problems with the Mueller protection bill. One is that to be constitutional, the special counsel must be an inferior officer. Otherwise, he has to be confirmed by the Senate, which Mueller wasn’t. And if he’s an inferior officer, he can fired.
Mueller can’t be an inferior officer in some respects and a hypersuperior officer in others, enjoying protections from his ouster that even Cabinet officials don’t enjoy.
The Mueller protection bill would really represent a return to the constitutional anomaly of the old independent counsel statute. There is a Supreme Court decision that hasn’t been directly overruled, Morrison v. Olson, upholding that law.
As Amar notes, though, the decision’s credibility is in tatters.