I just beat that final driving mission. Before which the closest I got was 2.x seconds behind the lead car a couple of times.
I won in dramatic style: I lost control under braking at high speed in the final S turns, spun out and hit the wall hard going backwards just behind the lead car in the 90 degree turn before the pit entrance. Fortunately, the game doesn't penalize hits on the BACK end of the car, just the front! So I had enough time to compose myself and chase the leader through the last 2 corners, beating him to the finish by half a second.