And you'd have to show that your ducks are poultry viz " domestic varieties of duck" not merely ducks that you have as ornament but ducks which common sense and the law sees as 'poultry''
.Furthermore you can only shoot the dog if it is in the act of "worrying livestock". and as a last resort (you have to act with reasonable cause, you see). You can't just shoot it because it's there and obviously you can't shoot it because of what it's done in the past.You'd have to have poultry there at the time and it has to be "worrying" the birds, that is biting them or plainly , and patently, on the point of doing so.That'll be for you to show (and, believe me, that is not as easy at it might sound, because the law is more on the side of the dog and than yours!)
You'll lose your legal protection , such as it is, if you don't report the shooting to the police within 48 hours.So you'll have the police involved, whether you like it or not, and have to satisfy them if you're to avoid prosecution for criminal damage, or any civil action.
It's as well to heed the cautionary warning in the Shooting Times piece [link]. It's lot easier and a a great deal more sensible, if at all possible, to take more measures to protect your birds by other means.
You're very ill-advised to try it (note the warning in the Shooting Times piece}.