Care hardening using case-knit might be a very good idea.

Replacing it with a piece of good spring stock or drill rod, properly hardened and tempered would also work - perhaps the best 'fix'. Sounds to me the firing pin sitting in the spent primer is what is locking up the action.

imho, .004" is about perfect for headspace.
I also think that primers backing out only .001" is very good on a single shot - and perhaps shows only .001" headspace on 'that' case.