it makes me laugh when people come on with this kind of indignant outrage, asking for help yet refuse to take any unless it means they can do whatever they want.
you have been given plenty of advice but all you can say is 'why shoud l, why should?' .... you should because you have a problem which is fairly easy to solve.
what you cant do is just go around shooting everything and anything that moves just because you feel like it, and if you are expecting someone here to say yes you can do that, then you will have a long wait.
if you are concerned about the cost of chicken wire, then what do think its going to cost to go through the courts when someone sues you for killing their dog?
even a camo net would do, that may be cheaper.
get a sign up first off and see if that has any effect - if not, get a net, get dog deterrent gadget that emits a high pitched tone, and put down deterrent powder.