Set traps and get a gun to finish the ****** off, if you don't have the heart to garrotte him.
You'll get a fine if you get caught. Well worth it.
I've dealt with neighbors who let their dogs run. Generally, telling the owners you're setting traps and when the dog gets trapped, you're throwing it, living or dead, into their yard, takes care of the problem. Sorry your pup had to die first. I managed to intimidate my neighbors into controlling their dogs before they actually killed any of my livestock.
Though they did kill our sheep, bite a girl in the head, decimate a herd of sheep.
I think the asswipe who owned the dogs killed my Great Pyrenees...who never left my yard acreage. He got pissed off because my dog got him cornered when he was sneaking around my house, and that's the last time I saw my dog. Who wasn't even half grown.
Some years earlier the guy across the river had two bear dogs who would routinely cross the river while their owner laughed. The dogs each got one end of my cow dog and pulled, and nearly killed him, before I showed up and chased them off.
I loaded my dog, injured hideously in the neck and internally (he had a hernia that never went away, wheezed in the kitchen for a week, and was never the same) into the back of my pickup, drove to the neighbor's house, a rifle on the gun rack, and told him I was setting traps along the perimeter of our property, along the river, and if his dogs came anywhere near my goats or my dogs again and I was out there, they were getting shot. He said "But that's ILLEGAL! aND...W-w-well your little dog runs back and forth along the banks and ANTAGONIZES them."
I told him if my dog ever crossed the river he could shoot him.
His ******* dogs never crossed the river again.