I don’t know that a vet could do much and I wouldn’t worry about one piece. But it would be a good idea to pick up all the twine you can when you’re feeding this winter. Years back I saw a cow that died and hardware was suspected. They drug her off and we checked her after the buzzards and coyotes had done their work, just to see if we could find what she swallowed. She had a ball of twine in her that was almost as big as a basketball.
We’ve always picked up wire and twine just to keep a place neat, but that showed me there is another reason. You’ll never get it all but it’s a good habit to gather all you can. Most of them will try to eat around it, but obviously “there’s always one.”
I agree with Craig. We routinely "police" our areas for new trash. Usually after a rain, new pieces of metal, wire, glass turn up. Doesn't take that long to walk a grid through pen, sub-pasture areas--also gives you a peaceful walk to check fences, cattle, etc. I always find something that could be hazardous to an animal, however small it is. "Ounce of prevention is worth a pound of cure."