I personally would cull. I cannot imagine the pain. It will be severely handicapped for the rest of it's life as well.
I get that we think we know what a chicken will have to live with after losing a leg, physiologically and physically speaking, but here is a little something for you to think about before you cull a chicken who has lost a leg, wing, etc., to a predator:
1. This is not (probably) a genetic reason to cull, unless you think inexperience is a good reason to cull (FYI, I do not).
2. A chicken who still has the will to live after a severe injury is a stronger than average chicken (FYI, I do believe this).
3. A chicken who can survive a severe injury is a stronger than average chicken (FYI, I do believe this).
4. The will to survive and propagate is one of the best survival instincts. I definitely do believe this.
If a chicken shows the will to live by eating and drinking, I personally do not cull. In my flock I have individuals who have survived predator attacks who have gone in to win or to produce winning stock with champion status at shows. That is not just my opinion, but a trained poultry judge's opinion.
That said, a little supportive therapy, (that is, a quiet box with food and water to recover in), will tell you if you have a survivor. IMHO, NO ONE IS REQUIRED TO CULL/KILL A CHICKEN. We can choose to do that because we have the power. No one including God (check the 10 commandments, the Koran, etc.) requires us to cull on any basis. That is inherently a personal and a private decision.
My advice is to do the best you can for anyone or any animal under your care. That is all you can do, and no one has any legitimate basis for challenging your judgement. Just do your best.