There must be a way, but I'm not aware of the formula. Perhaps ChatGPT can come up with one? It would also have to take into account the masses of any spokes and hub used to support and rotate the ring. Those masses and their shapes would affect the moment of inertia of the flywheel.
Edit...
If the transducer's resonant frequency is affected by the cleaning load you may have to tweak the oscillator frequency accordingly to maintain maximum amplitude.
If the cam (not the gearing) provides the force to lift the weight and you want the weight to fall freely (with the gearing disengaged) then the gearing serves no useful purpose that I can see.
A problem I see with your gearing is that as the weight rises it will cause (or assist) the flywheel's anti-clockwise rotation, but that continued rotation will prevent the weight from subsequently descending.
So the flywheel drives the cam, the cam drives the scissor mech, the mech moves the weight, the weight moves the gears, the gears rotate the flywheel, the flywheel drives the cam, ........ I hope you're not expecting perpetual motion?
Re-thinking the maximum torque needed, it won't occur at the maximum radius of the cam as assumed so far; it will actually be at the minimum radius by my reckoning, for the following reason.
You want the weight to rise at constant velocity; but when the weight falls to its lowest point it...