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Just open the snare loop. No reason to cut the cable.
Never saw the thread but I have the Knipex 8" bolt cutters, they are handy. They have the leverage of a blot cutter but are more of a beefed up wire cutter. Don't let the term "bolt cutter" through you off.
When these guys are hunting with their dogs, if a dog gets in a snare, it stops and waits to be let off the leash. It’s easy. I would call your situation a free roaming dog issue. I’ve never seen a dog twist a snare cable at all. Also I’ve caught a couple lost hounds, and one lost chukar dog, and returned them to their owners. I was called by a musher. He’d had a dog get off the chain and he couldn’t catch it. It was eating a dead deer along the rr tracks and he was afraid it was going to get hit by the train. I set foothold traps and had it in a couple days. No problems.As I said, just open up the loop and let your dog out. Or open the trap and let your dog out. No sense in spending more time and energy destroying another’s gear.