So I'm limited to 8.33ms (1/120) shutter which is good enough. Oddly, if I dropped the shutter speed to 4.165 (half of 1/120 which should be on the 60Hz frequency), I still had black bars.
That is quite simple: Your lamps are pulsating in brightness 120 times a second. If you take pictures with a shutter speed of 1/120, you will expose for the duration of a full brightness circle. If you set the exposure time anywhere less, a picture will not take a full brightness circle to expose but lose some of it. So some frames/lines get more and some get less light.
If you want to play with it, drop shutter speed slowly below or above 1/120. You may find the bands speed or width may change.
What can you do? I don't think you can do much at the camera side. Going below 1/120 will give you an effect like this. Some of the temporal smothers may help reduce it, but as it is a reality of the world, the camera will notice it.
But there is an alternative option you have. Remove the pulsation at the source. Your lights are powered by AC current, or is there a 12V PSU in between? If there is, you can exchange it for a pure DC power supply (one without ripple). That should stop the pulsation and hence the end the problem. (In theory, unless in your lamps, new ripple is introduced by additional transformations)