Interesting debate, but for me, there are a couple of pieces of information missing from the story.
1. What is the proximity of the backyard in question to the house where the kid was thought to be missing? If they share a property line or are a few property lines away then I absolutely would want any searchers to trespass to find a 3 year old kid.
2. What is the history of the neighborhood with the dog and owner? If the dog was known to be aggressive and there was the potential that the kid had wandered into its territory, then again, I would hope the police would move quickly to intervene.
3. Did the mom report any concerns about her child encountering the dog?
If none of these are in play and the officer ended up encountering the dog in an area that it wasn't likely for the kid to be, then was surprised when the dog became aggressive and had to take drastic measures for his own safety, then he was wrong and should make amends.
If we lived in Japan, the police officer would personally appear before the dog's owner and formally apologize. The chief of the police department would also apologize and would pay restitution in an amount equal to the cost of a new dog. The owner would accept these actions and the issue would be closed.
In the USA, the owner will end up suing the department, accusing the officer of illegal trespass and misuse of a fire arm causing the death of his beloved dog. The officer will counter that he was legally bound to look for the child and the owner was at fault for harboring a dangerous and unpredictable beast. The court will find for one or the other, resulting in either no money for a new dog, or 10 times the amount of the cost of a dog. Both sides will incur court costs that will be passed on to the tax payers. Regardless of the outcome, neither side will really be satisfied, both sides will continue to distrust each other, perhaps even more than before and we'll continue to live in a divided community.