If there is any possibility of change in the design of the system, that would be the preferred route. I used to work for an OEM in the paper industry and we sometime sold an oscillating unwind option. We used a simple hydraulic cylinder to move the entire unwind stand back and forth between two limit switches. Speed control was by adjusting flow controls. Of course, every winder we sold had a hydraulic power unit anyway, so we only added the incremental cost of the cylinder, valving and plumbing to the system. If your system doesn't already have hydraulics, the approach would probably cost too much. I get the sense that it's bigger than you could expect to do with a pneumatic cylinder.
Another thing for you to consider is that once you take on this project you my well assume total ownership of the problem in management's eyes. If your solution doesn't live up to expectations, they will forget that the root of the problem is in the mechanical design of the system. That fact that it doesn't work will become entirely your fault.
That being said, adding a VFD and and an analog channel from the PLC probably won't be a huge dollar cost. Given the torque available at the output of a 3600:1 gearbox, I don't expect we're talking about a huge motor here so the cost of the drive isn't going to break the bank. If you have an appropriately sized drive sitting on the shelf, it becomes a project you can try out without a huge amount of risk. If you can't achieve enough of an improvement, put the drive back on the shelf and move on to the next project. Who knows, you might even be able to rig up a linkage off the cam to adjust a speed pot and not even have to use the PLC at all. Wouldn't that be sweet!
