Tie to baling twine - it snaps by itself. So long as you use a double loop it won't snap unless the horse is panicking properly.
Regarding leading, there are very few cases where I'd let go of a lead when a horse pulled. Really quick way to teach them a dangerous bad habit. If they're the sort that throws the head up or pulls back a jerk or two on a rope halter will get them behaving. Otherwise IMHO so long as it's not actually dangerous, hang on (making sure of course that the lead isn't looped around your hand as it never should be). Cotton ropes generally don't burn too bad. A horse that knows he can pull away is a danger to himself and others, not to mention very annoying.
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Yeah.. I don't see any reason why baling twine wouldn't work. I've seen it used for tying in trailers, how do you use it for tying everywhere else?