Here is schematic for muting a two wire microphone. The in's and out's can be swapped. You can treat the arrangement as if it is just a normal two wire mic. That means that it will have to be hooked up to a transmitter or other bias supply in the normal manner. There will be a faint click due to RF in area from the transmitter but it won’t be objectionable for your application. This only requires a single pole, single throw switch. When the switch (or push button) is closed the capacitor is connected to the two wires and shorts out the audio. There will be a little audio bleed-through, mostly at low frequencies. The larger the capacitor, the less bleed there will be. The resistor across the switch is necessary to keep the capacitor lightly connected to the bias supply so there won’t be a large pop when the switch is closed. A better but much more expensive solution is to use a unit designed to be silently muted by a DC control signal.