Precision with a continous servo motor
I have worked on a airplane simulator that also had this problem.  We used a stepper motor, which I think is also a good answer for your case.

A stepper works well here since you know it will need to deliver very little torque and therefore can be counted on to not loose absolute position once it has been established.  I don't know why you think a stepper motor will be more expensive than any other small motor, but I don't think that's true, especially when adding the cost of position feedback and gearing.  One thing you do need with the stepper is add some way to know you are at a fixed position.  A opto-interruptor is usually a good solution for that.  At startup, you hunt for the known position, then the firmware tracks it after that.

A servo-controlled motor will be more complicated because you need to add position feedback somehow.  That could be a pot with two wipers, or more likely, a shaft encoder.  The bare ordinary (as apposed to stepper) motor will need to be geared down, so that adds more cost.  This just sounds like more trouble to get working and to keep working, and I doubt there will be a cost savings after you consider all the other stuff this method will need beyond the bare motor.