Time for a bit of math.
Is the signal Stereo or Mono? This doubles the amount of bits calculated by Coding Badly.
Just another thought:
Imagine an Arduino handling the button in the Main studio (just add a relay or so). Let that same Arduino have an ethenetshield and a built in ethernet/web server that provides a virtual button. This virtual button can be "pressed" by another Arduino that also has an ethernetshield and client software and an real switch.
switch@remote) ---->[arduino remote] ====> (over the internet) =====> [Arduino Main] ---->relay ----> mute switch.
You could add intelligence that after every press of switch@remote the relay will restart or ignore it. In a second release I would add a check [e.g. LED] so that [Arduino remote] can see if [Arduino main] is alive & kicking ...
Is this a valid alternative?