optimal low pass by trial and error?
I would direct drive the mechanism coil with a PWM of a frequency that is well above the audio range and at least 20X the mechanical response time. The PWM driver chip that you have now may just be able to do that and if not find one that can do it. The coil will do a wonderful job of reducing the PWM to its equivalent analog value just the same as any PWM direct motor drive works.

You will want to have a clean PWM waveform driven by counter/timer hardware or by a dedicated PWM generater chip. (There are actually simple I2C generator chips available that you could interface to the Pi). A software generated PWM will likely have too many glitches and interrupt update latencies to produce a clean waveform.

A really nice thing about this approach is that you can change the PWM to any setting you want and get corresponding action. Then you can consider an algorithm in software that sets the PWM farther that its target value to get the mechanism moving and then taper off to the target value to settle to final position. With a little fiddling you should be able to tune the performance of the overdrive without any real feedback as long as the drive voltage and mechanical load stay the same.