Just a couple of ideas:
1) An IR LED (or array) on the back of the lead vehicle; the following car tracks it for steering, and detects its brightness for distance information. Use IR because the sensor can have a filter to exclude ambient light.
2) An ultrasonic emitter on the lead vehicle, direction and distance detected as above.
(However, I understand that dogs can hear ultrasound, so they may start chasing it as well.)
The exponential response may just be a result of the root mean square law, that a radiated signal is not received at a directly proportional strength to its distance. But this could be an advantage, because a very weak signal could be used to elicit a catch-up-quickly acceleration response, with a stronger signal reducing speed as it gets nearer.
