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.
Sorry, but my bird dog is more valuable than your snare, if she gets caught, the snare gets cut. Snares kill bird dogs every season, hence the subject being brought up. I carry bicycle cable cutters made by Park Tools.