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Is that a Breatlhess shifter?
At any rate if you can't drill & use an easy out in it's present location/position, you will need to pull the shifter out & do the work on the work bench.
Your talking about the 4 bolts that hold the shifter assembly to the shifter's box. You only need to remove the shifter & then the 3 bolts that hold the box in place.
You should be able to remove the box and do the job on the bench.
Either by using an ez out or drilling & tapping a hole for the next larger sized bolt.
You really need to purchase a set of different sized ez outs & drill bits.
How much of the broken bolt is exposed? I have often used a file to make a slot in the bolt and use a screw driver to back the bolt out.
That bolt shouldn't be bottomed out in the hole so it should be fairly easy to extract. You could still carefully drill the broken bolt in the very center and look for an easy out. They sell them there but you would have to figure out where. Order some on line.
How much of the broken bolt is exposed? I have often used a file to make a slot in the bolt and use a screw driver to back the bolt out.
That bolt shouldn't be bottomed out in the hole so it should be fairly easy to extract. You could still carefully drill the broken bolt in the very center and look for an easy out. They sell them there but you would have to figure out where. Order some on line.
BC
the bolt is a nail under the surface,, i will try to search for such a product locally,,