When I energize my motor with the 36v4 tic controller, it jerks and vibrates my setup quite strongly. I recognize it needs to lock into position with the magnets, but is there any way to reduce this vibration and make it less violent? Here is a link to a video of it:
That looks like fairly normal startup behavior. It looks like the movement of the stepper motor shaft is fairly small and does not vibrate (it just shifts into position), and the large inertial load of your system is what is causing the vibration. There might be some way you can mechanically dampen it to reduce the oscillations after it moves (possibly some kind of rotary damper), but I do not have any specific recommendations for that.
As far as reducing the movement of the stepper motor shaft, having a shutdown sequence that moves the motor to a full-step position might help a little, unless the load on the shaft causes it to turn when it gets de-energized. You could also try setting the current limit really low when you energize the stepper motor and ramping it up to your desired current limit to see if that smooths it out at all (though I am not sure how effective that will be).