sourland gave you some good advice. Invisible fencing, and a shock collar, with training are a good solution.
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You said he knows basic obedience, so he is smart and trainable. This is a good sign. For any new skill start without any distraction. Maybe pick a quiet room in your home. Grab a lawn chair and an old towel and lots of treats and maybe a leash depending how things go. Lay the towel on the floor next to the chair and take a seat and wait. Any time your dog comes and stands on the towel reward him, pet him, give him a treat. You must stay calm and seated the whole time. Eventually your dog will learn you want him to be there by your side. Once he gets this figured out do not reward him until he is sitting next to you on the towel. Once he gets this figured out do not reward him until he is laying down next to you on the towel. This could take a few weeks. A training session is done when your dog is bored and no longer engaged, maybe 10-15 minute sessions 3 times a day. Your dog is learning to be calm and still and the towel and lawn chair anre now visual cues to help achieve this desired behavior. Next move. Do this is a new room of the house. Maybe a familiar person strolls in and out of the picture. Eventually move outside. Pick a part of the yard with the least distraction. Eventually move to where the chicken run is visible at a distance, eventually sit in the lawn chair and spread out the towel next to the run fence. One day you will not need the chair or the towel or to be present with your dog. It will take time and consistency. It will not be perfect. Chickens making sudden movements will likely always be exciting. Hopefully your dog will be less excited and see that you do not find the chickens exciting and will not need to bark constantly to alert you to the boring chickens.
sourland gave you some good advice. Invisible fencing, and a shock collar, with training are a good solution.
I didn't even know that the beagle could hunt mice until one night when she did it. She was an owner surrender because she could no longer hunt and became gun shy all of a sudden (according to owner). They basically had her living outside for a few years because she couldn't hunt so it has taken a lot of training to make her a house dog. She now sleeps in the bed and gets fresh eggs for breakfast... I believe once they have that hunting instinct you can't break them of it. I know you can give dogs melatonin to calm them down, I give it to the beagle in thunderstorms. Maybe offering treats when he doesn't act a fool will make him realize you don't like when he's barking and such.I wish Patrick could catch the chippies like yours catches mice! We have so many of the little buggers around here, we could do with a few less.
All in all it's not a big deal, I would just love for him to be chill with it all.
Now if it would only stop raining so I could sit outside & work on this with him...