try getting a 80-400 with a TC2.0?
...and hope he doesn't encounter an over-enthusiastic security guard.
TS, I usually just shoot with a 70-200 for F1. You can get shots where you can see the eyes, but you'll need to do it while the cars are stationary for the start, then do some cropping. Panning along the start/stop straight during the race will be very very difficult if you are on high focal lengths to fill the frame with the car and driver, but it's much easier at wider angles.