Hello, ive have quite a few dealings with dogs worrying sheep and the fist thought is to grab a rifle, after all its the best tool for the job! But there has never been and never will be a firearms certificate granted for the shooting of dogs specifically, I imagine you could argue protection of livestock but I wouldn't bank on it.
If you're going to do it and have no choice I'd advise getting close enough for 42 grams of BBs to do the job (and not in a fac classed shotgun)
Petty I know but it only takes the wrong kind of police.... person, and you've lost your rifle.
On a second point its probably best in this world of social media Village life to be totally discrete, better to hear people talking of a missing dog than an evil farmer!
Best of luck with it, it's a horrible thing to deal with.