I'm with Cayugad on this one as well unless you know a competent machine shop better than any gunsmith. What needs to be done is to have the barrel put in a vice under a drill press with a very small reverse drill bit. It can be done with a regular bit and a very small easyout.

On my preicous Encore barrel I would leave it to the professionals.

One other thought sometimes with broke off screws they are not bottomed out and you can unscrew them using a pin or something like that.