I don't think he should go to jail, but from all appearances this was reckless endangerment and incompetence, and he killed a member of someones family who did nothing wrong. Maybe not a human member, thankfully, but a member nonetheless. He did significant harm to an innocent human - the emotional bonds people can have with their dogs can be pretty intense. He fucked up, and bad.
I don't think he should go to prison, even though most people would in the same situation, but I DO think he should lose his job, or at least be moved to permanent desk duty and lose his right to a weapon, preferably with a pay cut.
Again, let me be clear - he fucked up bad, and hurt an innocent family who did nothing wrong, potentially very badly, because he approached the situation incorrectly. And personal experience indicates cops who do things like this are dangerous to let out on the streets. Today, its killing a dog. Tomorrow, it's letting a criminal take your shotgun and go for a joyride in your police car because you have no bloody fucking idea how to do your job. The day after that, who knows.
None of this ruins his life - this is all perfectly in keeping with how society treats fuckups. If anything, its treating him with kid gloves. If there was any ruining here, it was on his end - discharging your weapon in the field is always, as a police officer, an extremely serious matter.