It's not a question of Constitutionality but pragmatism. The press is always looking to destroy Republican by an means they can, they sure don't need any help.
The press will violently and falsely attack the VP no matter what he does. It is just stupid for you to suggest that he should alter his behavior to accomodate the leftist media.
And even worse to suggest that he step down. Use your head, man.
YOU - HOO!
HELLO!
VP Cheney has not given them any help. The local press was notified. It's the WH press and MSM who are rabid and mad because they were SCOOPED.