Noob here, so take my comments as such.
I quieted by old Diana Model 27 somewhat by adding neoprene washers at either end of the spring, to prevent metal to meal contact and dampen harshness and vibration a bit. I'm told the spring torques and de-torques as it travels, so maybe adding a thin stainless steel washer between the spring and neoprene washer would be a good idea, too. Could be, the cocking lever is being contacted by the piston as it moves forward, so a small rubber bumper here might help.
Car stereo shops use Dynamat and generic equivalents to quiet cars. This has foil, asphalt, and adhesive backing. It's ~1/8" thick. Maybe some strips of this stuff, judiciously added to the inner surfaces of the stock or hidden outer surfaces of the gun (where otherwise covered by the stock) would help. Noise happens when hard materials like metal or wood vibrate. That's how percussion instruments work. The Dynamat stuff dampens vibration, and therefore noise. Just a speculative thought, without benefit of testing.