[TheOtherOne]

Two questions: does the present situation qualify as a rebellion or invasion, and which branch of government is thereby empowered to suspend the privilege?

I think there is 1 question only: "does the present situation qualify as a rebellion or invasion"?

As for the other,

Article I Section 1. All legislative powers herein granted shall be vested in a Congress of the United States, which shall consist of a Senate and House of Representatives.

I would think therefore that Section 9 is considered a power granted to Congress.