Could driving a stick shift aggravate my lower back injury?

Yes: It could but so could the act of driving as sitting creates a force of 4 x your body weight across the back and if in a vehicle, that creates a vibration of 5 hertz, which is the same frequency that a human spine will vibrate at placing a harmonic frequency stress on a 4x loaded back. That is why you feel stiff after a long car ride with no breaks, without a lower back injury.