While working on mounting our motor in the paper towel dispenser body, we decided it would make the most sense to connect it using a belt and pulleys. This prevents us from having to drill into the paper towel dispenser's plastic axle to fit the motor.

We got this belt from Amazon:

https://www.amazon.com/dp/B098QH4MW3

and are using this design as the base for our pulleys:
https://cad.onshape.com/documents/cf7b858fb3c2f64bb9c06e22/

We're going to 3D print the gears and some of the required components for adapting it.

We expect to do a few revisions after the first print, but we're optimistic about the approach.