Considering you're gonna spend 3+ months on the "cycling bit in the middle" , an extra 4 days riding from Washington to Yorktown would be an obvious solution, and it'd be my choice.

In fact, my wife and I did that a few months ago. The ride was absolutely beautiful in the springtime, there's a wealth of history in between DC and Yorktown, the roads are good, and it proved to be a good (relatively flat) 4-day warmup for the "bit in the middle".

There's a "Cycling Route 1" that runs down the east coast (the TransAm is "Cycling Route 76"). A portion of Route 1 goes through Washington to Ashland/Richmond; from there you can veer out to Yorktown or just turn west.

Check here for Virginia's state bike map.

Three drawbacks to this plan:
1) Finding a cycling route from the airport to DC. But that's trivial compared to researching the public transportation options that you're doing now.

2) Once you get into DC, you'll want to spend a week or so sightseeing..... Kind of blows a hole in your schedule

3) Leaving DC in the morning, the first day has very limited options for overnight accomodations. On Route 1, the only camping between Occoquan and Fredericksburg is off-route about 15 miles; some motels, but they're very expensive. But there are a lot of parks where it should be easy for a cyclist to pitch a tent.

-- Mark