So May 31st? If so that's the weekend after Memorial Day which is better than the weekend before, less crowded.We are talking the last Saturday of May using the Metro to go into the city.
It should be an easy drive on a Saturday morning from your campsite to Shady Grove metro, assuming you get an early start like 8. Traffic in that area probably won't be a challenge early on a Saturday morning. Parking is free at Metro stations on weekends and you shouldn't have problems finding parking, at least in the morning. Once the Metro train gets going in Shady Grove, it would be 45-60 minutes to Smithsonian, depending on how long you have to wait on the train change at Metro Center. Add in time to drive to the metro station from the campground and possibly wait for the metro train at Shady Grove (they normally run every 15-20 minutes on the weekend) you're probably talking a minimum of 90 minutes to get in....and it could take you 2 hours if you take a wrong turn, take your time, etc.
At Shady Grove you are catching the Red Line. The easiest approach imo would be to go into Metro Center metro stop and change to the Orange or Blue line (Orange line going towards New Carrollton, Blue towards Largo Town Center). Go 2 stops and get off at the Smithsonian. That puts you right on the mall at the end closest to the monuments. It's still gonna be a hike but imo this is easiest way in given that you're coming in on the Red Line. The other option is to connect at Metro Center on the Orange or Blue going the other direction (Orange towards Vienna, Blue towards Franconia-Springfield) go 3 stops and get off at GWU/Foggy Bottom as the other poster suggested. I think the walk from Smithsonian stop is more interesting but the other may be a little shorter (we never have used the Foggy Bottom station).
From 1 + 2 use Smithsonian, change to Red line at Metro Center and go to the end aka Shady Grove. That's another reason to possibly get off at Smithsonian on your way into town, may be a little less confusing to use same station both ways. Just be aware the Metro Center can get real busy if there's a game at Verizon Center that's getting out just as you arrive at the station. That of course assume the Wizards and/or Capitals are still in the playoff and playing at home that day, both of which are uh iffyAfter lunch we will give families three options:
1. Air and Space Museum
2. Natural History Museum
3. Lincoln Theater & House Tour
........
Which metro stop do we use to return to camp?
.Not sure about option 3 - do you mean Lincoln Theater up on U Street or Ford Theater where Lincoln was shot?
HTH
. Each driver will be provided a copy of the directions. That doesn't mean they will be able to follow them. Sadly, its a reality. Car pooling will be done by the very few who only have two members going. It is mostly being attended by entire families, though, and they will need the extra room for tents, sleeping bags, etc. 

The average temps in March are in the 50s I believe, so barring a polar vortex again.... But you just never know.