Hi, did you get anywhere with the suggestions?
I've also been developing line followers using the Pololu 10:1 (standard) micro gear motors. Boy, these motors are quick! I've also needed to use negative motor speed values to get around 15cm(6") radius bends. I've noticed similar behavior on bends to your vehicle, even though I'm using the 32 x 7 mm wheels & tires.
I found the 'skidding' was caused by having too great a negative speed value on the inside motor - the amount of applied twist exceeds the friction between tire and track on sharp bends.
I also found that there was a balance between PID coefficients and the amount of negative speed applied. With more negative speed, the P & D coefficients can be reduced.
Of course, we say 'negative speeds' but in practice, I do not think the inside motor actually reverses, just applies additional braking.
Hope this helps!