I'm not following how treason would prevent the court from hearing the case… That sounds kind of silly.
You couldn't prove treason without a case in the first place.
. . . Congress actually hears charges of treason.
However, the court case didn't allege treason. Simply violating constitutional procedure doesn't count as treason and the case for treason would likely come from discovery during the prosecution of a case which turned up evidence.
Now, if we were in a state of martial law of some variety, then yes - the military could prosecute treason by tribunal, but this would come only at the inability of Congress to hear the case of treason or at their delegation (and the delegation of that authority may be challenged in the supreme court - meaning that if the national government is functional, congress will likely be decided to not be able to delegate its responsibility to hear treason cases).
Schitt has no say in it as the Senate brings the charges forward. I would have to review the process, some, but the Constitution does not recognize senate or house committees. Those are fan fiction rules they all came up with. The senate prosecutes the case of treason and the House is the jury.
Of course, the problem comes down to, as well, that we have a party seating majority offices whose members are on trial in some cases.
The States will effectively have to drive all election results into dispute. This will also coincide with a vetoed funding bill.