I would still suggest a stepping motor, this will give you the precision you need and as the load is light there should not be any worry about needing positional feedback, you can just use dead reckoning.
The only difficulty is in determining a reference position for this system. There have been many suggestions for this, however as you have the physical system then you can determine what would be the easiest way to do this.
It could be an optical slot switch, a micro switch or even a proximity switch sensing something about the thrust of the capacitor core.
The only thing you omitted is the speed you want to change this tuning at. However, I suspect this is slow given the application.