If you have to wait 3 weeks to leave you will never make the ride in time. Days too short, weather too iffy. If you left now you would spend the first couple of weeks getting in shape while you travel and just might make it.

An alternate suggestion is above. Check into Amtrak and how your travel route can intersect with a big Amtrak station. Maybe Birmingham, AL or Atlanta, GA. It has to be a major station that deals with checked baggage, because you will have box and check your bike. Amtrak has boxes available.

You could also start out on Amtrak and go to New Orleans and ride from there.

I wouldn't plan on getting on the Amtrak Southern Crescent(New Orleans to DC) the day before the event. I would say that the Southern Crescent from Atlanta to DC will be booked the night before going north and the night after going south. It is a night trip both ways and way to go to the inauguration and not have to spend the night(except on the train).