I just wrapped a loop of HME rubber wire around between the cams, similar to your bungee. I think it's thicker than your bungee, but the rubber coating makes it nearly noiseless. No need to take apart the housing.
The key is being able to use 2 hands to remove the rope from the cleat as you detach it from the tree. One hand pulls one of the fingers of the cleat open the other pulls the rope out. That’s how I keep my Muddy stick cleats quiet. Building a one-stick at some point...and will employ the same method to keep the cleat from making noise
