I'm of the opinion that Ward did sight and shoot Buddy from a distance. He fired the gun to give the dog a fighting chance to escape, and used the sniper rifle to track and shoot the dog. (They showed Ward using the rifle earlier.) They didn't, however, show if he actually *hit* the dog, but that's immaterial to the point about his conflicted character they're showing....Ward's "weakness" means he can't/won't shoot someone defenseless in front of him, but to convince himself to follow orders without the baggage, gives them a fighting chance to survive.
That's given some weight when you consider how his "execution" of FitzSimmons went down. He could've outright killed them both in hand-to-hand, but didn't. Instead, he gave them a fighting chance to survive by ejecting them from the plane. In his mind, he carried out his orders without having to do it in cold blood. It's splitting hairs, but to someone like Ward, that hair becomes a lifeline to sanity.
I'm really liking what they've done with his character, and cannot wait to see how it all plays out.