Obviously you do not understand trapping. There is no such thing as a "leg snare" that I know of A snare is made to go around the neck and kill in very short time. A leg hold trap is a tried and true method of trapping predators. I am willing to concede the induce some pain he animal, but the predator will induce pain into the chickens if they are not stopped.
Relocating an animal that has a taste for farm animals. like chickens, is simply moving the problem to someone else. You have not solved the problem. I would never "relocate" a predator, I prefer to take care of my own problems and dirty work and not have to p to someone else and their flock.
Free range is a great way to raise chickens. Predators re terrorist and should not be allowed to continue their terror.
Now, I would try dogs and locking birds up at night, but a fox that wants to eat chicken will, bringing their own demise down on them.
The only reason to use a live trap is to hold the animal, until can kill it.