It definitely is possible, but only just.
I did it with Racing Medium tyres, and fully tuned except for some of the weight reduction because I ran out of money. Default setup.
You can usually get to 1st place before Turn 1 with the Veyron's great acceleration, and after that, its just down to not making any mistakes, and defending where necessary. This can be very hard, particularly on the Nordschleife, as one small mistake can lose you the race.
Cape Ring was the hardest for me, as I didn't know the track as well as I knew the others, so I made some errors. I finished 6th on that in the championship, and then it took me several goes to get 1st when I went back afterwards.
Anyway, I ended up equal 1st in the championship (equal with a Mazda 787B) with results of 1st 1st 2nd 6th 2nd.