I see the Senate being maintained in Federal control due to the loss of almost all territory east of the Mississippi. Those 26 states are elected by "courts-in-exile", the descendants of the politicos from those states who were in Washington and any refugees who might have fled. While the holders of other state capitals still send senators, these Westron senate families keep things form getting totally out of hand, and of course generate some insane political intrigue.
As to the house... possibly due to the state of emergency since elections can only be reliably held in federally controlled territory, all of the House is for Federal territory? Less sure on this one.
You could then elevate that state of emergency as adding a lot of quirks. You could also have the state of emergency take on a religious signifcance, since it will only be ended when the Union is restored and thus tied in to Millenarian ideas?
I think Presidents should be elected by all of the states east of the Mississippi, plus all of the courts-in-exile And since a Census can not be conducted on the western states, they maintain their pre-regression electoral votes. And if there is some bumfuckery and no one gets a majority, it gets thrown to the House which (assuming my prior theory hold) is in American hands. Possibly the state-of-emergency makes these for life, or failing that 10 years.
I believe that while there would be literal political "dynasties" descended from old presidents, generally speaking the convention is for Presidents to assume the name of a former President. Possibly this began with some President who saved the US from total dissolution and thus gained the nickname "George Washington II." Further Presidents received nicknames based on their similarity to old presidents (especially as the Church becomes increasingly obsessed with 'classicism'). Eventually to control the narrative Presidents start pushign the names themselves.