I don't think there is a solution with PWM and gearmotors. The issue is that they don't start moving until about a 15% pulse. There's nothing to be done about it.
I took a servo lens apart to see how they do it. Servo lenses have excellent range of control. The don't used PWM. They drive the motor with straight DC and vary the voltage. Contrary to the name servo-lens, they don't use servo motors, they are simple DC motors.
I don't know of a joystick motor controller that outputs a variable DC voltage. The ones I know of all output PWM.
Servo motors don't use PWM either, they use PCM. That said, I don't see how a Servo Lens could do it with just a DC motor, there is no way it would know its current position.
Anyways, the best solution would probably be to go belt driven from a servo, not a gear motor (which is a DC motor w/ a gearbox). With a rubber belt, you will have a small amount of elasticity which should absorb the initial jerk. Just like the rubber band/tripod trick for smooth panning.