American Liar noted

I hadn't paid attention to the Chris Kyle story until the controversy over American Sniper, so I am a little surprised to learn about how he has been shown up as a self aggrandising liar, at least with respect to his exploits in America after returning from Iraq.

I wonder if the exposure of this, shall we say, problematic aspect of his character led Spielberg to drop the project?  Or did it only gain ground after he was shot?

Update:  from a Slate movie critic reflecting on the movie:

The falsehoods in American Sniper are dangerous because a lot of audiences leave the theater thinking that Chris Kyle was a role model. I’ve actually gotten emails from military vets who were also troubled by the film. A lot of them are even harsher on Kyle than I’m comfortable being, in part because I’ve never served and in part because I was once attacked by Glenn Beck’s online army after poking holes in Lone Survivor. But American Sniper convinces viewers that Chris Kyle is what heroism looks like: a great guy who shoots a lot of people and doesn’t think twice about it. Watching American Sniper, I kept wondering who Kyle himself had been imitating. Sylvester Stallone? John Wayne? Or the ultimate irony, Clint Eastwood himself as Dirty Harry?