How do I get my bench clamp swivel to hold?
Your idea of grinding a few threads from the bolt seems to be on the right track. In that photo, I wondered if the bolt was bottoming in the handle nut. Of course, if you grind too much off, it's not an easy task to replace the lost threads.

As an alternative, consider a shim of sufficient diameter to engage the entire bottom plate of the vise with sufficient thickness to take up the needed gap in the clinch bolt assembly.

If you're handy with a welder, build up a bead of weld on the edges of the bolt that engage the slot and gain the take-up in that manner as another option.