So I'm going about this all wrong? creating the tapered part of the halyard is not done by extracting the core from the cover in the area to be tapered, it's done by "sliding" the cover off the core to expose the core on one end (creating the taper.
I think it can be done both ways, but pulling the core out shouldn't be to much trouble once you learn it.
Robbie D showed me what he uses for his spin halyards. He pulls out the core and then tucks the cover back into the core. i did this and it was a bit tricky. it required tapering the jacket (and a lock stitch).