The separation-of-powers doctrine does not require federal courts to stay all private actions against the President until he leaves office. Even accepting the unique importance of the Presidency in the constitutional scheme, it does not follow that that doctrine would be violated by allowing this action to proceed. The doctrine provides a self-executing safeguard against the encroachment or aggrandizement of one of the three co-equal branches of Government at the expense of another. ... But in this case there is no suggestion that the Federal Judiciary is being asked to perform any function that might in some way be described as "executive.'' Respondent is merely asking the courts to exercise their core Article III jurisdiction to decide cases and controversies, and, whatever the outcome, there is no possibility that the decision here will curtail the scope of the Executive Branch's official powers. The Court rejects petitioner's contention that this case-as well as the potential additional litigation that an affirmance of the Eighth Circuit's judgment might spawn-may place unacceptable burdens on the President that will hamper the performance of his official duties. That assertion finds little support either in history, as evidenced by the paucity of suits against sitting Presidents for their private actions, or in the relatively narrow compass of the issues raised in this particular case. Of greater significance, it is settled that the Judiciary may severely burden the Executive Branch by reviewing the legality of the President's official conduct ... It must follow that the federal courts have power to determine the legality of the President's unofficial conduct. The reasons for rejecting a categorical rule requiring federal courts to stay private actions during the President's term apply as well to a rule that would, in petitioner's words, require a stay "in all but the most exceptional cases.''
...The Court is not persuaded of the seriousness of the alleged risks that this decision will generate a large volume of politically motivated harassing and frivolous litigation and that national security concerns might prevent the President from explaining a legitimate need for a continuance, and has confidence in the ability of federal judges to deal with both concerns. If Congress deems it appropriate to afford the President stronger protection, it may respond with legislation.
https://www.law.cornell.edu/supremecourt/text/520/681
That case involved an incident that allegedly occurred before Clinton became president, while this one occurred during Trump's presidency - he made a false statement about Carroll after she claimed he raped her before he became president. I don't see the difference, though, since under this analysis there would be no greater burden on the executive branch if the president has to defend against a claim involving an unofficial act while he was president than against one that occurred previously.
Barr is probably going to try to rely on
Nixon v. Fitzgerald, in which Fitzgerald was dismissed from his job during a departmental reorganization and reduction in force, in which his job was eliminated. He alleged that his separation represented unlawful retaliation for his congressional testimony and named a number of officials, including Nixon, as defendants. The Supreme Court held that a president is immune from civil actions based on official acts:
... petitioner, as a former President of the United States, is entitled to absolute immunity from damages liability predicated on his official acts. We consider this immunity a functionally mandated incident of the President's unique office, rooted in the constitutional tradition of the separation of powers and supported by our history. Justice Story's analysis remains persuasive:
"There are . . . incidental powers, belonging to the executive department, which are necessarily implied from the nature of the functions, which are confided to it. Among these, must necessarily be included the power to perform them . . . . The president cannot, therefore, be liable to arrest, imprisonment, or detention, while he is in the discharge of the duties of his office; and for this purpose his person must be deemed, in civil cases at least, to possess an official inviolability."
https://www.law.cornell.edu/supremecourt/text/457/731
I will be very interested in seeing how Barr tries to explain how calling someone a liar while president because they accused him of raping her before he was president constitutes an official act of a president.
= new reply since forum marked as read
Ahhhh, they don't care. They'd throw themselves in front of a speeding train just so Hair Furor's hair wouldn't get messed up. Stupid is as stupid does. 
