The leadscrew play discussed above I am assuming is longitudinal Y axis play back and forth? This would seem to cause problems threading all the time. When you reverse with a tumbler, it could cause a slight mismatch as you switch directions with the gears. This might account for it happening intermittently with repeated passes, as this will setup a mismatch between the chuck and the leadscrew.
You can also make the brass tipped set screws quite easily, get some brass rod, trim off the tip of the Allen screw and insert the brass rod with some Loctite. I have some fine thread 3/8-24 x 1.25" Allen set screws that I could fit with tips and send you, if you want to try to use these to lock your chuck. It is also less likely for your chuck to unscrew using a slower VFD acceleration.
That is what I use all the time to snug up on threads when needed. When I have needed one right away I have even drilled the end of cup point setscrew and pressed in a piece of brass.
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