Wow, these Nutria sound like damn fine fun, although the cottonmouth snakes could put a spanner in the works.
Don't know if you are allowed to "spotlight" in the US, but it is a good way to grab a critters attention at night. Rabbits stand and won't move much at all in a spotlight so I'm wondrering if these nutria would do the same....make 'em easier (and cheaper in the long run) to shoot.
Feral eradication can be a great fun sport, and most importantly, you aren't doing any harm to the native fauna...if anything you are helping it.
I know that some say that spotlighting isn't very sporting, but when you are trying to eradicate a pest I don't see that it makes any difference....in fact I rather like the idea of the M79 loaded with 40mm HE grenades. A fowling piece is ok...but I reckon a punt gun would be better....the term overkill doesn't exist in my vocabulary (well not where pests are concerned).
Seriously though, had a mate here in Australia many years ago who used to take a short barrel single shot .22 out with 6 .22 rimfire shorts and come back with 6 head shot rabbits.
From the way you have described these critters maybe this is an idea you could pursue....damn cheap to shoot and no noise to speak of, and a real challenge.
Snakes.....got shed loads of them here in Australia. The cottonmouth sounds a bit like our Tiger snake or Taipan. Both of these nasty buggers will have a go just for the fun of it, and they are ranked as the most poison snakes in the world. Know people that say the only good snake is a dead snake....don't like killing them myself though.
Good hunting and do everything you can to help eradicate this pest....we have a similar problem here in Australia with the Cane Toad...one very ugly toad that is responsible for a lot of destruction of native animals.