Easy to say that now Andy, but there must have been more to it, and there was, in his mid. I feel he may have had a void to fill in his conscious after suffering the demotion. Honor in family name comes to mind. Imagine having to go home a private and having to answer questions about why you're not a sergeant anymore. That could be an earth shattering event. Perhaps he felt this action would elevate him in status and overshadow the demotion. Or maybe it was that he had an idea to be the last act in the Great War, kind of a "take that!" to Pershing and anyone else involved in his demotion.
Could be any number of reasons why he did it.