I think that staging a coup would be treason.
The US Constitution, Article 3, Section 3 states, "Treason against the United States, shall consist only in levying War against them, or in adhering to their Enemies, giving them Aid and Comfort. No Person shall be convicted of Treason unless on the Testimony of two Witnesses to the same overt Act, or on Confession in open Court."
A coup is essentially war on a small scale; a conviction might be possible.
IF the coup actually happens, and the plotters get their hands on DC (ASB level success, or a very long term, meticulous plan where everything goes perfectly) I suspect that it would be the army that resolves the situation. It could be the navy, though; it was marines under the army's Colonel Lee that captured John Brown in '59.
Since some southern governors were willing to use their troops to oppose integration (Little Rock Nine) might they, in a more volatile situation, send their national guard, or elements of it, to DC, in response to the "president" calling for them?
Obviously, things have been going from bad to worse compared to OTL for a while for something this extreme to happen.