Good day. I am not a pilot but I am a broadcasting engineer. Excuse my interloping. Your problem was simple to diagnose and is somewhat easy and inexpensive to resolve and is something we deal with every day.
The attenuator and ALC in your camera or any other consumer or prosumer camera may not be sufficient to counteract the sound power levels required to deal with the noisy environment of a cockpit. For example, the power needed to listen comfortably in your home is less than needed in your car. So the earphone amplifier in the comm system has been designed to provide all the gain needed for that environment.
I will make the assumption that to avoid missing an ATC transmission the volume level of aircraft comm radios cannot go to zero. Even at minimum that appears be too much for the ALC and attenuator in your camera.
I heard two things in your recording. Distortion: the comm was over driving the input to the camera. And background hiss. This hiss is thermal noise in the comm system's headphone amplifier and is only objectionable when over amplified above background levels as demonstrated in the recording.
The suggestion vapilot2004 made is on point. If you have access to a technician in an avionics shoppe they can cobble together a 15 or 20 dB pad from spare parts in their tool box. Except for the special aircraft connectors, the rest are readily available at an electronics hobby shoppe. I Googled this commercially available device for about 14 Pounds from
http://www.canford.co.uk 
Place it in line between the camera and the wiring harness.
A final comment about ANR headsets. The magic you hear is all in your head! There are microphones inside each ear cup. The circuitry analyzes what it hears inside the cup and creates an audio signal 180* out of phase. When mixed with what is coming up the wire, leaves only the radio or intercom. The squelch action you describe is just the ANR readjusting to blessed silence.
Good luck,
Bob Sudock/Fox Television Engineering (retired)
Los Angeles