The only objects which experience accelerations are the cars (during the second phase), But nothing at all is accelerating during the third phase. But in the second and third phases both cars are experiencing propulsive forces.
It is entirely possible that I have overlooked something in this analysis, and if you can find it and point it out to me I would be grateful
What you have overlooked is the momentum change of the slipstream of each car (ie the source of the drag). The one is opposed to the other, hence you have two masses (or m-dot in this case) experiencing opposite momentum change and overall conservation of momentum of the entire 'system'. Whereas, with a single vehicle there is momentum change of the single slipstream and opposing momentum change of the Earth resulting from the net torque on the road surface. Where there is propulsive force there are equal and opposite momentum changes and conservation of momentum.
F = m.a = mΔv/t = rate of change of momentum