Yes, the more you twist the rubber band the more torque is produced.
I’m not exactly sure how your bottle and rubber band are configured but it sounds like the same principle as the rubber band powered airplane toy.
The rubber band is connected to the propeller at one end and anchored at the other end.. You turn the propeller with your finger which stretches the rubber band storing elastic potential energy in the rubber band. In turning the prop you apply a torque. The force you need to apply increases as you stretch the rubber band much like the force needed to stretch a spring increases the more you stretch it (Hooke’ law). You release the prop and the rubber band applies a torque to the prop converting its elastic potential energy to rotational kinetic energy of the prop.
I think a thinner rubber band might limit the amount of stored elastic potential energy simply because it might be more fragile and break, but i’m Not sure.
Hope this helps.