Sunday Morning rotary motion challenge

What you are looking at in this video is the mechanism from a CA 1970's computer paper tape punch machine. (the hall sensor and blue plastic parts weren't on the original mechanism - I added them)

https://youtu.be/lagO27NlFHo

It originally ran at a fixed speed (much faster than this obviously) attached to a small standard AC motor.

What I'd like to do is to make it variable speed. I happened to have a BLS-510 from Stepperonline in house driving an inrunner BLDC motor (they came as a set). The motor has plenty of power to drive the load, but as you can see from the video the problem is the load is different based on where in the cycle the mechanism is and the feedback loop doesn't keep up. I tried putzing around with the PID settings in the control software, but I don't think PID is really fit to deal with slow loads where one part of a single rotation is more difficult than the other part. That and my changes only made it worse.

Then I got to thinking about flywheels - sort of a mechanical capacitor - to overcome the point of highest load. But it would likely need to be geared up and quite heavy and that feels too complicated.

I considered using a stepper motor, but while it would push through the load without much trouble I don't think it would provide the top speed I am likely to be after.

So I thought I'd consult a large group of electro-mechanical geniuses and here we are.

Let me know if you have any clever thoughts on how you would approach this problem, or if there is a solution out there that I don't know about.

Thank you,

DougM