You can't park there overnight, but if you go in AFTER midnight, you get to park until the next evening.
Take Metro to Union Station. Walk to the Capitol. Pay your respects. Return to Union Station. Catch Metro to return to the station where you parked.
You probably ought to sleep it off right there at the station for a while. Then, grab a meal and drive back.
The odds are against you finding a room in the WashDC metro area at this time!
I don't know that we'll be able to find a room, if not, your plan is the next best alternative.
Thanks for the warnings and options!