Grab a second Mini and tune it just enough to win that first race. There's an engine swap you might consider later on.
If you're trying to win the race with the chilies, you'll need nitrous for that.
Nitrous doesn't add PP.
One can add downforce by lowering the front and raising the tail, this doesn't add PP.
The fully customizable manual gearbox adds less than the racing gearbox. Adjust it so the car redlines in the highest gear towards the end of the longer straights.
Also look into the different differentials, in some cases the more expensive parts sometimes will reduce PP.
Adding the stiffness upgrade reduces PP.
Widening the car sometimes reduces PP, as does changing rims and aero parts.
By the way, if you widen the car, buying a new car body will not change it. Widebody is the only true permanent mod. If you swap the engine and then buy a new engine, it'll be the stock variant of whatever you drove the car up with, not what originally came with the car. That means if you install the Civic engine and over tune it, when you buy a new engine you'll get the Civic engine, not the Cooper one.
Best advice for the first race, buy another '65 Cooper, add nitrous, and done. When you want to do the Clubman Cup, run it over and over til you win, using what prize money you do get to buy parts. Stay on the road and do NOT hit the AI cars.
