Does your neighbor know you killed his dog and what do you expect his reaction to be? That would be my primary concern at this point in time.
In Texas he would be well within the law shooting the dog. The key being it had been caught in the act killing his chickens. There has to be threat or damage involved, you can’t just shoot one because it’s on your property.
If this happened where I live, that neighbor better be concerned about how much damage the judge is going to award me for the cost of my chickens.