Getting the rest of the screw out is a challenge, or maybe not.

My approach (if tools available at reasonable cost)
Build up the existing 1/8 length with a micro TIG
Epoxy the head to a piece of steel so it can be chucked
GENTLY machine the weld area to size
Thread the new stub to suit.

May sound hokey, but I have done similar.

Getting a new screw made and engraved may be cheaper, dunno

Dumb, but learning...Prof Em, BSc(ME), CAE (FYI)