Would L'Enfant be ok if I don't have a scanner?
My guess is that they are not to hot on plain-old-loitering, but it would certainly draw less attention in the first place, so it's probably a fairly safe try.
Also, I was thinking, about 25 years ago, I had a meeting in a building in which the FAA was renting some space at the southwest end of Maryland Avenue (time flies!). There's a circle there with the tracks emerging from underneath, just as they go across the Tidal Basin and the Potomac: maybe not the best place, but doable. Also, I've taken walks around East Potomac Park (just to the south of the Jefferson Memorial; orientation-wise, just across the river from Washington National Airport) and the tracks go over the trail. I've seen a few trains from there; again, not the best place -- you're kind-of looking up at the train -- but quite doable.