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I think you have your routes/descriptions reversed, but the first list of highways (I-81/I-40, etc), that would take you through VA and TN, is about 100 miles shorter and doesn't involve any time on 2 lane highways, so that's almost certainly going to be faster - barring things like weather or a severe traffic incident.

This is still a 2,000 mile trip and the fastest it can be done safely is with 3 overnight stops, and yes that is even with multiple drivers.

While there certainly speed patrols, there's really no such thing as a speed trip on the interstates because speed limits are almost always nearly constant - unlike some 2 lane roads where you can have speed limits unexpectedly reduced for no apparent reason except to generate revenue.

Speeding to any significant degree on a trip like this if often counter productive, as anything over 5-10 mph is pretty much an invitation to get pulled over. Get stopped once and any time you "saved" by speeding pretty much gets thrown out the window. Driving much beyond that also requires more work, effort, and attention as a driver so when do stop, you usually end up needing longer rest breaks anyway.