Quote Originally Posted by Peter Laidlericon View Post
ragged series of drilled holes around the edge and break it out with tthe snipe nosed pliers. It might be messy and a bit time consuming but you've saved your butt, some money and you'll still end up with a perfect fix. After all, the drill-out oversize, glue and plug is what Armourers do to stripped threads every day
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.