You are certainly in the right, and everyone knows it. To say you wounded or maimed the poor dog, well that was only because you failed to kill it. You did your duty, and your objective was to stop the threat and protect what is yours. Last time I checked that is perfectly legal in this country. Actually, this may suprise all of you:
In the eyes of the law, injuring economically valuable livestock is traditionally a more serious matter than injuring a person. In at least one state, Minnesota, a dog owner is even guilty of a minor criminal offense - a petty misdemeanor - if the dog kills or pursues domestic livestock.
The two cardinal rules, which apply almost everywhere, are:
1. A livestock owner is free to kill a dog that is killing, wounding, chasing, worrying, harassing, or attacking livestock.
2. A dog's owner or keeper is financially liable for any livestock damage the dog causes.
You're going to get the minority trying to scare you that you shouldn't have done that, blah blah. As far as the law is concerned, a pet is a pet, livestock is livestock. Emotions mean zero in this case, so stick to the facts. This is a not a bleeding heart case. The dog trespassed and did harm to your property and livestock. No brainer case closed. However, you said something about not being able to shoot a gun "in your city". That may be a cause for concern if your city does not allow livestock and many do not. Including fowl, but I digress.
So, here is where you pull the old reliable state laws from the "olden days". Even here is Michigan, there are REAL tough laws about dogs killing, molesting, chasing, etc. Specifically, the "Dog Laws of 1919", Section 287.280, states in part
..a township officer or other person designated by the township board shall request the district court judge to immediately issue a summons against the owner commanding him or her to appear before the township supervisor or township officer or other person designated by the township board and show cause why the dog should not be killed.
and states further...
Upon the return day fixed in the summons the township supervisor or township officer or other person designated by the township board shall proceed to determine whether the loss or damage to the livestock was caused by the dog, and if so he or she shall immediately notify the sheriff or the animal control officer of the county of that fact and upon notification the sheriff or the animal control officer shall kill the dog wherever found. Any owner or keeper of the dog or dogs shall be liable to the county in a civil action for all damages and costs paid by the county on any claim as provided in this section.
And, if you don't want to have to go through all of that malarkey, you can just handle the problem yourself next time:
Michigan State statue Dog Law of 1919, Act 339 of 1919. Section 287.279 spells it out very clearly in Section 19
Any person including a law enforcement officer may kill any dog which he sees in the act of pursuing, worrying, or wounding any livestock or poultry or attacking persons, and there shall be no liability on such person in damages or otherwise, for such killing. Any dog that enters any field or enclosure which is owned by or leased by a person producing livestock or poultry, outside of a city, unaccompanied by his owner or his owner's agent, shall constitute a trespass, and the owner shall be liable in damages.
I am sure if you look hard enough you will find such laws in about any state. So, if the city/township does not provide for it, you can cite state law. And, if you're lucky enough to live in Michigan, I just did all the work for you!
Oh, by the way, if the Sheriff does not take your report, which it is not the Sheriff but most likely a deputy, then call the Sheriff him or herself, whom is an elected official by the way, and file a complaint about filing your complaint and tell that you will file it with the state police next. Don't be afraid to stand up and be heard. If they don't want to help, then ask them if you can have their written permission to handle it yourself next time.
Good Luck.