I don’t think it’s fake. I’ve read the arguments about whether it’s fake or not... one of them saying that the pup doesn’t move. But in the first second of the video, you can see the dog lick its upper lip. It is limp because that is the natural instinct of a puppy when held by the scruff of the neck.
Second, it does make a “thud” sound when it hit the ground... I don’t think a stuffed animal would have done that.
That being said... this should not reflect on the Corps, and those left-wing kooks are just using this to promote their idiotic anti-war agenda.