As a retired Army senior NCO, I will say that the Gunner's actions are not all that uncommon. The military (I'm painting with a very broad brush here) makes a lot of noise about integrity, honesty, ethics, etc. And then everyone (broad brush, once again) lies, cheats, and steals. Yes, definitely report the incident and the Gunner's lack of integrity. If he falsified documents in this instance, what else has he lied about? He has absolutely zero credibility for anything at this point, and every action or decision that he has made needs to be reviewed, starting with his first day on the job. Expecting all of the cadets to lie to cover his lie is despicable.
Just be prepared to live with the consequences of your actions, whatever they may be. There WILL be retribution of some sort.