He could complain because your use of a firearm annoys him - however:
a) never shoot in the direction of his boundary thus preventing endangering him / his property.
b) Use a suppressed 22lr with subsonic ammo - this will be extremely quiet so that removes noise complaints, don't be flashy with it or set up a plinking target near his boundary so he's less likely to see it.
He could still complain that it scares or annoys him - however you have taken all relevant precautions with regards to safety, and mitigating any effect (noise) on him - if he complains from there it should be thrown out as it is a normal activity in a rural setting, pest control is an essential part of rural living. Hell he wouldn't get on to well in my area - especially in a couple of weeks when a lot of the ponds around us will sound like a warzone.
note - I wouldnt recommend a 22lr for goats, pretty inhumane with such a small margin for error on shot placement.
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