If there is loss of livestock I'd be willing to bet you could get a go ahead nod from your local game warden. You can hunt coyotes at night here in Missouri, just can't use a light. Which is completely stupid. I'd be willing to bet that if a Missouri landowner could prove he'd just lost five calves to coyotes (which seems a really high number btw) the landowner would be allowed to use a light at night. I know one of the local conservation agents I've discussed this with in the past has a Foxpro and night vision rig to assist landowners with just such a situation. Your best bet (like most of the questions in this forum) is to have a discussion about this with your local game warden.
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Every normal man must be tempted, at times, to spit on his hands, hoist the black flag, and begin slitting throats.