That was a bad deal, starting with people ( stupid people video taping cute wild animals in the hood ), ending with probably a dead dog, that had no interest in a fight, and no protection from its owners. I would have shot the doe, and probably would have gotten away with it here where I live, however, from the video, the opportunity was the slightest of windows, you would have to be right there, see the imminent danger and react, that dog was done for in under 20 seconds. To me it would have been no different than defending your pet from a pit bull, or coyote, sad for the other side, but your pet is your responsibility, both as an aggressor or to defend.
2 stories, LE friend of mine shot a doe with a Ruger .44mag carbine, going away shot, she fell down, he went to retrieve and she got up, he shot again, and she fell down, long story short, as he approached her, she was still breathing, he pulled out his knife to cut her throat, well she had different ideas and they fought, he did win, but told me, after 20yrs in Law Enforcement in one of the roughest areas of Dallas, that was the worse beating he ever took. Probably why when that happened to me, I had a Glock 21 in my strong hand, and when the deer popped up quick as lightning, I shot him 7 times, my partner in crime behind, said " I'm not sure I saw 7, but I definitely saw 6 cases in the air at the same time " Don't fight wild animals, they have a commitment we only dream of having to live.
Story 2, my Brother in law, was working on his car in George West TX, when a doe walked up on him and a friend, he grabbed a claw hammer and beat it to death, made the papers, unfortunately it turned out to be an escaped pet, he did clean it and eat it. No charges filed.