I know you say you've tried changing stiffness and additional force, but that does sound like the standard way to approach the problem you have. If that doesn't work with the sensitivity you need I expect you may be near too the robot limits of sensitivity. That said, it's worth considering exactly how best to set up your impedance mode.
So we can get better understanding of your process, how much force are you looking to apply?
Stiffness, damping and additional forces in an impedance control mode are applied in the direction of the tool coordinate frame, rather than base or world. Make sure you have configured the TCP orientation and set up your CartesianImpedanceControlMode correctly. As you want to follow the surface with a constant force, you don't want a significant contribution to the TCP force to be based on displacement from the 'hold' position, as such you want a very low Stiffness in the working direction. Damping is probably fine as the default. All of the work you want to be done by an AdditionalControlForce in the tool working direction.
It's worth considering the robot pose you are using during the operation. Near to any singularities the torque sensors become less accurate at calculating the TCP force. A quick way to do this is to set up a program with an endless while loop that prints the Force/Torque and ForceInaccuracy (ForceSensorData.getForceInaccuracy()), then jog the robot around. Limit your print outputs to every 500ms.
It might be worth spinning up your tool and seeing what effect that has on the recorded forces/torques. Gyroscopic effects and all sorts might be throwing off the force readings. Also, thinking about it, you must be running with some sort of powerchain up the robot flange - that is likely to be harming your force accuracy if it's strapped to the iiwa along it's length.
If you aren't able to get the results you need just using an impedance mode position hold, you could try mounting a roller/low friction surface on your tool and recording the positions that makes with the surface as the turntable rotates. You could use a higher force to ensure that surface contact is maintained. This would require either knowing the absolute position of the turntable, or being able to synchronise the movement. You could then actively drive the robot to match those positions.