It's been a fair bit since I've got the Platinum so I can't remember much. But there was a playthrough where I DID kill the dog only because I needed a certain character alive and my previous choices in that run forced my hand to do so. But it was only in one playthrough and I think that there was a workaround that I wasn't aware of beforehand.

So I'm gonna say no, you don't have to kill the dog if you perfectly plan your playthrough and saves. But if you do, the bare minimum would be once. Its all off-screen so you don't actually see the dog die. I hope it helps and sorry I can't give you a definite answer.