Sounds like the soft points are going up into a sharp edged hole at the bottom of the tapered recess in the seating ram that presses on the ogive. It swages the tip partly up into the hole, and that pulls it off. I would expect the lead to be stuck in the seater then, though.
Pictures would help. If I'm right about the mechanism, boring the hole in the bottom of the ram out wider than the soft tips would do the job. Simply sharpening a dowel rod and using it to spin some Flitz or other polish in there might help, too. Meanwhile, take your finger and put a little bit of case sizing lube on the soft points before seating, and see if the problem stops and if so, are there marks on the points where the hole is?
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