The cop may need some further training, but he should not lose his livelihood over what was a confusing situation and the death of a dog unless there is some egregious and definite breach of protocol. I don't care what kind of a dog it was. I think the issue is that he was called to a violent domestic disturbance and, in the course of trying to contend with a human he was worried about, disposed of the dog, which, for all he knew, could have undermined his ability to subdue a criminal.
If the cop broke some rule or protocol then I have no problem with him being censured per those rules of engagement. That should be obvious and need no real clarification. However, if the police feel he handled it adequately and no one can point to a specific violation, then I don't see a reason to take action against him.
I don't see any major issue here that would require a firing. I hardly see the need for admin leave and a major investigation. It should be fairly apparent on its face. Either he is allowed to brandish the weapon based on the info he had or he isn't. He didn't kill or obviously endanger anyone as his shot was true.
It is an unfortunate situation. Beyond that, unless some problem is specifically brought up in re rules of engagement, etc., I don't see making a big deal out of this.