Do you know the max torque required to turn the screw?
You can TRY something really simple. Get a DC motor/ gear train that is geared to be a reasonable time. You can brake a DC motor by putting a relay across the motor.
1. You can try just torque alone from a torque screwdriver and an electronic edge finder such as this one:
**broken link removed**
I still think the stepper and a small controller might be your best option. You could basically step quickly until you think your close and then change the step until the electronic edge finder finds the edge.
Proof of concept might be easy. A stepper driver and a variable frequency source and a gate. Just drive it really slow and see if it works.
Spring load the edge finder if it isn't already. You can either detect the led with a phototransistor or modify the device itself.
I seriously think, you can drive x steps quickly and then step slowly until the edge is detected.
Your SHIM could be made larger. Just have the right material either laser cut or etched.
You may not even need the SHIM if you could use a screw and a locknut.
As another idea,you might be able to start with the carcass of a cordless drill and retrofit it. Buy the case only when your ready to implement. The electronics could be separate and corded to the drill.