Apskip pointed out to me (correctly) that there's an alternate route from Hains Pt to the finish mat, via Highway 1 or Route 400 on the west side of the Potomac instead of I-295 on the east side. It's a toss-up on Google Maps, since the west bank routes are shorter, but are on smaller roads and have more lights. I've left the maps as they are until someone can verify if the episode shows teams travelling one way or another.
When I lived in the D.C. area, and worked downtown, I would usually use the 14th Street bridge and the George Washington Parkway when I lived south of Alexandria, which is literally on the way to Mount Vernon. (I also would park on the Virginia side and take the metro intown, and back at the end of the day, driving on the GW Parkway.)
In fact, it was a good thing I was doing that on the day a passenger flight leaving National airport crashed into the 14th Street bridge, and many people who drove to work all the way in never got home that night; the bridge was closed completely for weeks, and wasn't fully repaired for months. I was on the subway when that happened, and I only learned what happened after I got home.)
The issue would be what road construction would be going on; that often affects what routes would be taken. (The problem with going down I-395 to the Capitol Beltway is that you then have to cross over the Potomac on that bridge which has been kept open and rebuilt completely at least twice since I lived there, and the exits to the GW Parkway on the west side is not easy any longer.)
I suspect the taxis would have used the west side route and not the east side of the river from Hains Point to Mt. Vernon.