I haven’t seen American Sniper yet so damn you internet for putting this in my brain. From what I have been told the fake baby isn’t noticeable when you are watching the movie from the start, not just a clip.
So why was Bradly Cooper nominated for an Oscar? Well, instead of going the easy route and acting with a real baby, he made the concious choice of choosing a fake baby to deliver this intense performance. He not only breathes life into the Chris Kyle character, but also his child. Kudos, Mr. Cooper.