> unless the Government can show that applying a criminal prohibition to that act would pose no “dangers of intrusion on the authority and functions of the Executive Branch.” Fitzgerald, 457 U. S., at 754. Pp. 12–15.
This seems to be the real boundary. To show something doesn't have immunity you either have to:
1. Show it was an unofficial act
2. Show it wasn't part of the core presidential powers and that prosecuting it wouldn't have any danger of intruding intruding on "the authority and functions of the Executive Branch."
I think that last standard is related to presidential immunity from subpoena which courts have also recently construed quite broadly