The phrase "to shoot the dog" means to focus on or address a smaller or less important issue while ignoring or avoiding a bigger, more important problem. Full definition
Don't take people's stuff without good reason — and don't shoot their dogs while you're at it. (legalblogwatch.typepad.com)
This law is applied to dogs — in fact, state law allows any person to shoot a dog in pursuit of a wild animal under certain circumstances. (adn.com)
These professionals say that until fairly recently, this highly undesirable trait was kept out of the breed by shooting any dog on the spot that displayed aggression toward humans. (apamo.org)