Well, I should say relatively little money, you'll need ~$150k, but thats much better than the 20 mil racecars your competing agains.
I finally found the solution, like many people on here it is hard to find a car that can compete in this without having boat loads of money. I've finally found the solution, but you may not like it, as it makes use of B-Spec mode.
Winning the Nostalgia Super Car Cup in B-Spec will win you a Shelby Cobra '66. I forget what level you have to be to enter the Nostalgia Cup, but my B-Spec guy is only level 9, and he got to level 9 while completing that cup, so it can't be more than 8.
Tuning the cobra if I remember correctly gets me 731 hp, and a weight of only 916kg. Easily competitive with the race cars. You'll have to spend some time in B-Spec, but I'm level 21 in A-Spec and have won many races and special events, and this is the first car I've gotten that can really compete in this event.
I did it with the Chaparrel 2J Racecar. It can even match the Toyota 7, despite marginally lower speed, due to its SUPERB cornering. Not at ALL what you'd expect from a car that looks like a sloped aluminum box with two wheels.
How did I get it? Birthday Exploit '70. :-p The 2J and Toyota 7 are both from that year, so it usually doesn't take long to get one or the other, and you (or your B-Spec driver) can easily win with either. But yeah, even with the 2J, you do NOT want the Toyota 7 to be at the front of the field. >_<
Life is like a Blue Chest... you never know what you're gonna get.
You can also try using the '72 Countach you get as a prize car from the Gallardo only race. I haven't got the credits to upgrade it yet, but it feels like it will do better in that race than the upgraded Cobra does.