The photo doesn’t really show how deep the snag is. If it’s just superficial, in other words, about a 1/16 - 1/8” deep, and you can’t see cords/ body plies then the tire will run ok. However if you can see the body cords then I’d recommend replacing the tire.
Just remember, if you use it everyday or once a year, the potential for failure is there. I used to just shake my head when people would tell me “I only load the truck up once a year to get fire wood, so I don’t need a tire that is that heavy duty”. As I explained, that’s when you do the damage. Most opted to move to the higher carrying capacity, but those that didn’t always had a note written on their invoice stating they where told of potential problems by overloading. I only had one come back and argue the point, he walked out unhappy I wouldn’t warranty his problem.