Get a piece of dowel that just fits down the bore, and preferably a fairly long brass screw if you can. Drill the proper sized hole in the end of the dowel for size of acrew. Then screw the screw into the dowel, nice and snug. Dowel won't split if you predrilled it the right size, and didn't screw the screw in too far.
Use your preferred method of chopping of grinding of the head/unthreaded part of the screw. Then profile it back to a sharpened end, making it similar to the end you just screwed into the wooden dowel, don't grind or file away the threads. That's what is going to grab the bore snake.
Proceed to putting the dowel with the newly formed brass screw sticking out of it down the bore, screwing it into the fibers of the bore snake, then pull in the dowel and remove the broken bore snake. You can set up a makeshift slide hammer for easy extraction, if you use a long enough dowel, cross pin it on backside of the dowel and have a small weight or something handy laying around that will work, for the slide part of your makeshift slide hammer.
Very effective way of doing it without ruining your bore. It works best if you can pull it backwards, and not thread it in from the side that the rope broke off on. A metal screw can also be used in a pinch, but you are not "bore safe" then.