Im not sure being shot is worse than being rounded up and sent to slaughter, in fact for a feral animal I'm sure its preferable. And if they're going to be shot, why not charge money for the "pleasure"?
Or perhaps the goats should just be left to spread til they do huge environmental damage and then start dying of starvation?
Ideally of course you'd introduce wolves, though I'm not sure the goats would think that kinder (and nor would the local livestock or pets).