As has been said, a set of racing softs should be more than enough for you to take this race by a pretty large margin. (In fact, that seems to be the case for ALL of this round's challenges.)
If you'd like to spend a little less money, a few of the cheapest engine mods (ECM, converter, filter) along with Sports - Soft will do the trick as well though you've gotta work a little harder at it.
I'm sure a great driver could win this with no tuning on Sports - Soft. I'm able to consistently take 2nd with just the tires, but can't do any better than about 4.5 seconds behind. I swapped for Race - Soft and walked away with it.
The key, mentioned above, is smoothness and knowing where the racing line is giving you bad advice. I know it's very helpful when you don't know a track, but you should never consider it more than a general guideline. There are very often times where it's telling you to brake when you don't need to. This track, in this car, has several of them. Learn where you can just let off the throttle to get thru a corner instead of braking (almost every corner) and try to take as big and sweeping of a line as possible. With the lack of acceleration that this thing has you need to concentrate more on keeping your speed through the corners than taking the shortest route. (That's generally true of most racing lines, but it's even more important in this case.)