Except that:
(1) When doing live fire exercises, everything from missle prelaunch, to launch, to tracking the flight, to target acquisition, to hit (or miss) is recorded in multiple places on the ship and with the "umpires" who are running the exercise.
(2) The fire control team, the radar operators, and the bridge would know damn quick if that happened. And so would the umpires. And word would go through the crews mess like lightning. Heck, there are always extra guys hanging around - running coffee, manning the phones, etc. No way could they keep it quiet. Not possible.
(3) There is no "top secret" quick we screwed up pull out the "cover up" manual. The exercise would be terminated immediately, the umpires would be calling up one chain of command and the ship would be calling up another. Word would go out to the Coast Guard pretty damn quick. And the ship would immediately go charging out to see if there were any survivors at flank speed.
Nope. No way it was a live fire exercise mistake. That is tin hat stuff.