I got no ideas on how to make this better.

I did something similar on the pull starter for a lawn mower, the bolt was broken off and I also broke an EZ out in the screw.  It was a 6mm bolt and I was able to use a Dremel and chainsaw sharpening stones to grind away the EZ out and bolt remains enough to pick out the pieces...and a drill and tap restored the remaining threads in the engine block.  The small size of the mixture screw aren't going to make something like this work for you.

Sometimes those old carbs have really corroded threads that just refuse to allow the brass mixture screw to come out.  (I think having a small hole in the brass cover allows water to enter the area and cause the corrosion to the tiny threads).

I have seen quite a few stock carbs that had the outer threads crimped a bit by the factory, presumably to prevent the screw from vibrating out.
I have had to work the screw in & out, over and over before the screw would finally come out.  Others required the drill & tap, and still others drill and pick out the pieces like Dave mentioned.  Then of course there are a few that just went in the trash.
Yep, it’s a pain in the butt.

I do have carb kits if you decide to go that way, and I also have a stock carb that I think is in good shape but need to get a better look at it before I commit to that.