There is a military helicopter weapon launch scenario that does involve significant enough loss of weight to affect control positions:

Hover launch of a torpedo. Torpedoes weigh ~750 pounds. When that much weight comes off the aircraft, controls will need to be adjusted.

That being said, if a military helo is launching a torpedo in a hover over water, most likely it is a coupled hover, i.e., the autopilot is keeping the helo in a steady hover.

Even more interesting, is that the only time a helo is dropping a torpedo in a hover is if it is using a dipping sonar for targeting, meaning the pilot has to be careful not to drop the torpedo on top of the dipping sonar cable! The pilot will usually give the helo a small lateral drift towards the side of the aircraft that the torpedo is launched from to ensure the sonar cable isn't in the way. It has happened a few times where the pilot has mistaken which direction to drift and instead actually increased the odds of the torpedo striking the cable. At least one dipping sonar that I can remember was lost at sea due to an incident like that.

Immediately after launch, the autopilot will sense the upward motion and slightly lower the collective to compensate, causing the RPM to stay the same, but the engine output to decrease by a few percent.