Ive had a few of these happen. If you can get a clean hole in the top down close to the shaft contact point, then soak it with a few drops of WD40 or Breakfree with a few layers of paper towels underneath and let it set overnight. Then ...

- try first to twist it out using a small jewelers screwdriver. Might be loosened enough to spin out
- if not, then move to trying the screw extraction tool. There were about 20 different types on Amazon when I looked but, like you said you will need to find a smallish set
- if all else fails, ive had luck by drilling just one 16th smaller than original bolt/screw diameter, then using needle nose and tiny screwdriver to twist the remaing shell/hull removed most times without even damaging original threads.

Tough spot for it to happen but I'll bet you'll sort it successfully