Just found this thread...
I live between Annapolis and DC, and have spent alot of time in both cities. I worked an outside job in DC for many years, and know the place pretty well (or at least, the 1989 DC).
Wash DC has some horrible traffic. Anytime between Labor day and Easter the traffic is lower, but it is much cooler then. Around Easter is also the Cherry Blossom trees blooming and it's festival. The memorial areas you are talking about has these trees, in addition to the Tidal Basin and the Potomac Parks (the Jefferson Memorial is right at the Tidal basin).
As far as horrible traffic being a problem, I think the more bikes you have, the better to mitigate the effects of high traffic volumes. Personally, I would say to heck with it, and come just before Easter and see the cherry blooms. Depending on how many you get, it may be nice to check with a few hotels for group rates, as before Easter they are not usually booked up.