If that was me, I'd drill down the length of the remains, all around its edge until I'd virtually 'butchered' it out. Then simply bore the now 'butchered hole to a standard size (it's all covered by the butt plate anyway.........). Glue it al around internally and drive a hardwood plug in. Then using the butt plate as a guide, redrill and insert a new screw.
There may be another way I'm sure but I can't think of it..............