MOSFET, voltage divider and ADC
Before I go any further, you should consider using a microcontroller for this. I know this is a really simple circuit, but unless you know the signal source will never have short pulses you could end up losing a mute press somewhere and annoying the user. I'm assuming the function here is to mute the radio when another sound source is present. A microcontroller based solution can guarantee it waits the necessary time before pressing mute again. It would take maybe a dozen lines of code. You should get some ATTiny boards and familiarize yourself with them. You'll be glad you did.
A MOSFET will work just fine, be sure to drive the gate to a higher voltage since the circuit on the other side of the mosfet could be at +5V. Since you have access to the 12V supply it's easy to drive the gate to 12v which will ensure turn on.
Here's how to do that with a somewhat simpler edge detector circuit:
link

On the top left and right there's the switch module with resistors and the thing that reads it. In the lower left there's the new circuit that fakes a button press. It uses a 3.4K resistor (the value the switch module has when the mute button is pressed). The MOSFET acts as an electronic switch that can be turned on to switch the resistor in. I also included a simpler transistor based edge detector circuit. The capacitor can be changed to adjust the pulse time.
If you already have a logic circuit built or prefer to use logic chips to analog stuff, you can just wire the output of that to a MOSFET gate as in the circuit below and omit all the messy resistor and transistor stuff.
link

If you add another MOSFET then you can guarantee it will work regardless of the voltage on the other side. This is the circuit used in analog switch ICs albeit with back to back N and P channel transistors since those have to block voltage in both directions.