First please don't try this, as I'm not 100% sure it's safe yet. Still doing research on the components.
But it's simple, rather than putting your airpump on something, or in something, I put some filter foam inside my airpump to absorb the vibration. And I'm stunned at how well it works. It easily reduced the noise by half.
I was thinking about taking this a step further by perhaps using expanding ventilation foam (from the can) to fill in all the gaps completely. However that is more of a hard foam so I would still need some flexible foam to put around the moving parts.
My next idea is to buy a block of soft rubber, and make the housing out of that.
The sound is most likely coming diaphragm. You might need a new one.
The padding is only making the pump less efficient. It reduced the noise because it reduced the movement of the arm. It's not a bad way to reduce the amount of air going through.