Besides Cherry Hill (which is, by all accounts, a very nice campground and convenient to mass transit), there are a couple of other less expensive campground options to consider. The least expensive is probably Greenbelt National Park, which is not far from Cherry Hill. It's $16 per night, if it hasn't gone up since I stayed there a year and a bit ago, and is about a half hour to forty-five minute walk to the College Park metro station from the campground. This is typical national park camping: no hookups, but you have a picnic table and a fire ring and a shower house in the vicinity.
Another option is Lake Fairfax (county) park, which is within walking distance of the Wiehle-Reston East metro station. They offer electric hookups.
There are a great many museums to seen in DC beyond the Museum of the Bible (which, from what little I've heard of it, sounds quite interesting). Since that's a new museum, it may well be more crowded than it otherwise would be. I rather enjoyed the free docent-led tour of the National Building Museum, somewhat more so than the pay exhibits there. I also very much enjoyed the Air and Space museum extension at the Dulles Airport (which is an enormous extension--a full day to be sure). Getting to the airport with public transit requires taking a bus, at least for a few more years until the Metro extension there may be completed.