Going southbound all the way on South Cap from the Douglas Bridge to Overlook is possible (and legal), but requires a bit of a steady nerve. (there’s a narrow shoulder until just before the Malcolm X Ave entrance to Bolling, then a far right lane out of three that you can take entirely and most people will pass you unmolested.) I would not under any circumstances try to salmon here.
Alternatively if you’re going to NRL, I presume you can get on military bases? Enter the base at the Firth Stirling entrance (it’s open all through the day on weekdays), then go all the way on the base bike lanes/ sharrows (nearly completely flat the whole way) or the path on the riverfront back to this entrance http://goo.gl/maps/RNJQa which dumps you right on Overlook. (you can also go through housing – on a bike but not a car – to the next gate, across from Chesapeake St, but I don’t think that saves you much).
Alternatively alternatively, start from Alexandria, go over the Wilson Bridge, go up Oxon Hill and back down to the Oxon Run trail, then Shepards Parkway to the end, go under the freeway and you’re at the NRL entrance. http://goo.gl/maps/zd6CG Oxon hill is a longer and steeper hill than you hit going over and through the Congress Heights neighborhood, but zero traffic (and it’s all at one time).
ETA: If you’re coming from the north on the Green Line, and intend to end your commute by biking, I think it’s just as well that you end the Metro ride in downtown (somewhere between Mt Vernon Sq & L’Enfant Plaza) as it is to go across the river, because the Green line takes kinda a circuitous route after it leaves L’Enfant.
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