Drill exhaust manifold bolt - Remove head or not?

If it was me, I would remove the head and take it to a machine shop. And I'm not suggesting an engine machine shop but one that is HIGHLY skilled in one-off machining...hard to find these days. I'm not sure if you still have a bit of the extraction tool stuck but if so, they can use carbide tooling to machine straight through, leaving only the threads which can be picked out. I'm not a machinist but the machinist instructor where I used to work would do this on a regular basis and it always worked. Hand drilling will never be straight enough and while I agree the welding trick can work, it is always a crap shoot and the less you end up handing over to the machinist in the end, the more he will appreciate it.