so I will first attempt to get under the kin, then anneal if necessary.
You've got it bakerds. Anneal to soften, then cut.
When annealing, how can one limit the heat from affecting more of the screw than necessary ?
To my understanding, you can't. That's just the nature of the beast. Generally speaking, the only part of the material that will be annealed is the orange/red heated part, not the lesser heated parts so much. So you wouldn't really need to worry much. Get the torch on there and get the job done though. Don't fiddle fart around and let the heat go any further than necessary. Annealing usually involves letting the material cool at its own paces, in air. I'm not a master Jedi annealer, however, so corrections might be in order.
If it were me, I would cut the screw with a thin cut-off disk close to length. You know, those thin ones that you typically use on a die grinder.
Put a broken/dull carbide insert in the tool holder and get all the hard stuff out of the way. Yeah, it will break again, but who cares....it's already broke/dull. Another example of why NOT to trash all of your broken carbide.
Replace with good insert and finish it out. Ceramic if you have one. If not, no biggie.
I have also used the cut off disk, or grinder, to grind away the hard stuff. It's an easy and quick way to get to the soft stuff, and does not put excess heat into the shaft. Not very scientific, but it aint gottta be.
Since your just doin the one, skipping the annealing all together is what I would do. If you were doin quiet a few, it might make more sense to anneal before cutting.