That is one of the problems with DC, they don't care about people from out of town finding parking. They want everyone to take Metro.
Though, in this case it's federal property and not the 'city's property', though I can't figure out why they made the parking rules this way for these areas. That might be a good thing to bring up with your local members of Congress.
There isn't any other place to park near there. If you are driving, you are probably better off parking at one of the Metro parking facilities and then taking the Metro to the Smithsonian Metro and walking from there. Union Station has a large parking garage (the entrance is off of H St NE). Of course, the price to pay or parking anywhere in DC is usually not cheap.