Sunday cup should work , i have gold in those 3 races with bspec driver but there is at least 1 race series in Beginners class that says you need more than 1 driver
It's actually your b-spec level that determines the amount of drivers you can have. You get another one at level 10 as well. The driver's level is an independent thing. You can also race any of your drivers in any race you have available regardless of their level. This makes it possible to level up very quickly for new drivers. Just be careful with their stamina.