Whilst i hear your frustration, id play the long slow safe route along the lines of some if the points that Rod & RJ mention, such as video/trail camera footage & councils...
Im not sure how things operate up there, but id be especially cautious of using the excuse of the dogs attacking any stock to shoot them.
I just say this from experience, as a neighbouring farmer in an area i used shoot in, once accused my dog (a German Shephard) of killing his sheep...i knew my boy wouldn't & wasnt doing this.
Things got tense with many people involved, many sheep were killed, mauled & destroyed, i recieved many threats that my dog will be shot next time i come to the area, & i got 2 unfair fines from a council.
The farmer eventually took my (very stern) advice (or my counter threat seriously) & through his dilligence eventually discovered my dog was actually innocent & he ended up shooting 2 out of 3 other dogs that were the culprits...both the farmer & myself were thankful he didnt loose his cool on the trigger...!
Things would have got very messy if my dog was shot.
I understand your dilemma is different, but be careful-my point telling the story is you dont want to make a big mistake with a gun in a moment of frustration.
Do you have 'wandering dog' fines up there...?
If so trail cam footage with time & date should net him some losses to begin with....hell i got wandering dog fines from heresay that my dog was killing stock, so footage of his should do the trick...!
And a .22lr spitting sub HP's is much quieter than a .243...
(That cracks me up RJ-you put the terms "silence is golden" & ".243" in the same sentence...
)
Hope you get it sorted.
The man who knows everything, doesnt really know everything...he's just stopped learning...