With the tools at hand, I ended up using Peter's suggestion. We drilled holes around the broken screw until we were able to break it free, used needle nose pliars and twisted it out. Very little damage to the screw hole was done and the replacement screw is still able to get enough bite that it is tight enough to keep the buttplate secure. I will however fix it the correct way with a wood plug.
I appreciate all of the replies and suggestions. :clap: