PA Ave to Freedom Plaza; the old Council Bldg. I remember Ken Mease and others having a rally there to bring a team to DC. The Bullets had their parade in that area in '79. Centrally located and guaranteed to piss off commuters. Ideal
PA Avenue would be the traditional route. Like you said, maybe start at the Capitol and then end at Freedom Plaza. PA Avene, of course, is the route taken by the Inaugural Parade so they have a process in place for parades along the route. Constitution Avenue is also a favorite route. That's where the Cherry Blossom and St. Patrick's Day parades take place.
What might also work is a parade from the Capitol down South Capitol Street that ends at the park.
It's kind of fun just to think about it.