I want to comment on the idea of you carrying a gun. This is only a good idea if you are comfortable and competent. That being said, if you are legally allowed to shoot on your own property, perhaps setting up a target and target shooting at least once a week would be beneficial. This would help you become more comfortable with a firearm, but also potentially make your neighbors a bit more hesitant to walk onto your property unannounced. If they routinely hear gunshots from your property and know your schedule is one where you are often home, they'd probably think twice before wandering over.
In terms of a privacy fence, I know your neighbors are not exactly reasonable, but you could propose it to them and see if they will split the cost of the fence where it borders the two properties.
I know you shouldn't have to do any of these things, but a truce is definitely needed. What if you had an arrangement where occasionally an extra cockerel is culled and the neighbor gets the bird for dinner? It might be a way of keeping peace and they sound like the sort of neighbors that would be extremely satisfied to know the bird is no longer alive as opposed to knowing that you are trying to re-home it. I know you mentioned a breeding program so this may not work for you, but is something to consider.
In terms of a privacy fence, I know your neighbors are not exactly reasonable, but you could propose it to them and see if they will split the cost of the fence where it borders the two properties.
I know you shouldn't have to do any of these things, but a truce is definitely needed. What if you had an arrangement where occasionally an extra cockerel is culled and the neighbor gets the bird for dinner? It might be a way of keeping peace and they sound like the sort of neighbors that would be extremely satisfied to know the bird is no longer alive as opposed to knowing that you are trying to re-home it. I know you mentioned a breeding program so this may not work for you, but is something to consider.