Depends on how humane you're feeling.
If it's on the low end, you can drown it in a barrel or something like it, full of water and with a lid on. Or weigh it down and throw it in a suitable body of water. My grandfather used to trap stray cats in a cage trap, and he would just kick the trap over the edge of the quay, cat and all, and pull the trap out a few days later, when the cat would be drowned and the crabs and little fish would have started eating it. He'd shake the cat out, bait the trap and set it again. That's not very nice, and a terrible way to go. But it works, and you don't ruin the fur.
If you leave it in the trap, it's going to sort itself out somehow. That is an even worse way to go, and very cruel besides. But it works. I found grandpa's traps a few years ago, and three of them had the mummified remains of cats in them. Don't know when they got in there, but he died in 2002.
If you're feeling humane, hit it in the noggin as hard as you can with something heavy. A heavy metal shovel should do the trick. That should at the very least knock them out for a bit. Depending on the animal, the force might cause their eyes to pop out, or bulge out in a grotesque way. It looks almost cartoonish, and would probably be very funny if it wasn't so grotesque.
When they're knocked out, that's when you bleed them, by, simply put, cutting their throat with a sharp knife. Pocket knife, chef knife, don't make no nevermind so long as it's sharp.
Instead of a snare, you could use a striking trap, like a Conibear. I don't know about no nutria, but the mink-sized ones I've used are mean enough, so you'll want to take care you don't get your fingers crushed in it. Here's a video on how you can set them.
https://www.youtube.com/watch?v=v-8MJHLzR8EAnd make sure to anchor your trap properly, lest you have an injured animal running around. That's entirely unnecessary and easily preventable.
