The recommended tap drill size for a 6-32 thread is a #36. This is equal to 0.1065" or 2.71 mm. Small indeed for the welding, heating, bashing or wiggle, jiggle methods.
Can the hole location be moved? Can it be changed to a larger size? I have tried everything my limited brain power could come up with to solve similar situations. The two solutions that I have found that will work are EDM or making a new part.
The dull tap test is very effective, the dull one is the tap that just broke. Whenever I have tried the bashing method I have always had the same results. Both the 'basher' and the 'bashee' were damaged beyond use and a new part was needed. Perhaps flyingtractors has the proper cold chisel in his collection. Would that require a larger or smaller hammer? (OT, what is a flyingtractor?) The wiggle, jiggle tools belong in the same category as neck ties and high heel shoes. The hole size prevents the use of the left handed, self removing drills. A little online research will find some one time use EDM set ups that may help, such as the one that uses a door bell solenoid. Use this only when unwanted guests are expected. I have had no luck with chemicals as the errant tap is always of the wrong metal broken in the wrong metal. CH3CH2OH, in certain quantities, will help all of the mentioned processes.
I live in a rural area. There is a very rude black cat that travels my yard in the dark on occasion. He is not impressed with me even when threatened with extreme termination by use of explosive projectiles. The next time I break a tap I shall seek his assistance first.