I don't think the car is the issue. Racing hard tires have no grip in grass and if you go off track you have to be very very careful about steering, braking and throttle or else. Even on pavement with the traction control set to 5 you have to be careful with the throttle in the lower gears while turning else you will get wheel spin and the car will jump out from under you. It is much worse if you added that high end turbo which adds a lot of extra power at higher rpms.
As for which car, personally I like the Audi R8, it handles very well and is fast on all circuts but either of the ones in the OP should get the job done just fine. You just got to drive them and be careful