If there's a way to get out the broken bit with a suitably powerful magnet, that's got to be by far the approach that causes minimal further complication. Getting a replacement bleed screw for this brake may be pretty difficult (or Tektro may just have one to send, who knows). It is critical to use the right screw and o-ring so that the seal works.
Cow magnets are pretty strong and readily available, and the right shape. Of course a pickup magnet or computer magnet may be more readily on-hand. There may be some creative way you can figure out access to an electromagnet.
Drilling into the tool steel of the broken bit without destroying everything else does not seem very realistic to me.
In theory you could find something to bond a tool to the screw and broken bit, unscrew it, then use heat to undo the bond. In practice whenever I try something like this the bond doesn't really get strong enough.
Using an adhesive to pull on the broken bit is pretty likely to get it stuck in worse, although it's true that in theory this might be avoidable. You have the advantage that the broken parts probably want to mate closely and provide lots of surface area. You might try cleaning the surfaces with alcohol on a q-tip, put a very tiny dot of epoxy on the connection point, sticking it on and securing with a rubber band, and leaving it for its full cure time. If you can avoid using enough epoxy that any spills out the crack of the bit, you should be safe from making it worse this way.