When your driver starts out, let him do his thing but before he spins out, push the "pace down" command a couple of times (you have to time it right).
When his "temperament level goes down to about 3/4, you'll see he'll drive stupidly careful around that corner. As soon as he clears it, hit the pace up or maintain button.
And you had to post this because? If you don't like the mode, don't post in threads about it unless it's an "I hate B-Spec" thread.
To the OP, try using a lesser powered car. I forget what the typical range of HP is for the competitors but I ran the race in a 160 HP Eunos. Less power should keep him from spinning out and likely increase his lap times. Also go with the max tires you can, this can make a big difference.