I wonder why they don't just have an inert concrete mass on the other side and an exit port that opens into a sand bunker, sure it's a lot of energy but they don't have to contain it, only limit back splash from the mass hitting the sand. Doesn't even need to be enough sand to stop the mass completely, just enough to catch fragments, the mass can continue into the earth with no consequences

Yes, that's an intriguing idea.

What I came up with was this: the launch arm has a second mass inside it, that will move (via centrifugal force) away from the center, in the same moment the payload is released.

Of course, just having two payloads launch in opposite direction as you propose maybe has much less implementation problems (though I wonder how you stop that second dummy payload in a way that does not cause an earthquake or explosion :)

Sand will melt with all the energy absorbed. Water is much better.

You'd need to reclaim a lot of energy very quickly because after the projectile's exit the atmospheric pressure is going to slow you down right quick

You could slam a blast door closed behind the exiting projectile. It should be possible to place it in the exit tunnel so that it is fully closed before the inrushing atmosphere gets to it.

Plausible. Do you know how fast such doors can close?