I don't know anything about the new Sprinter, but I do know about the Audio-15 used in the NCV3:
What you propose is very difficult. There are layers of complexity that will conspire to thwart your efforts. It is not a simple matter of adding a switch or relay to control where the power comes from. First of all, modern OEM radios don't really have power switches. The Audio-15 is fed constant power, even when the "key" is off. Whether the radio is ON or OFF is decided by an ECU elsewhere in the vehicle. This permits the vehicle to be "smart" about turning the radio on/off when a door is opened, after being in park for a certain interval, and so on. When the ECU decides the radio should be ON, it sends a message via the CANbus (your vehicle's local area data network). The radio hears it and then decides whether to obey or not. As an anti-theft measure, the radio will go back to the CANbus and use it to determine your vehicle's VIN. If this succeeds, it compares the VIN to one programmed into it at the factory. Unless they match, you will simply get an error message on the screen. None of this will happen properly even if the radio is properly powered, since the ECU will be able to see that the vehicle is not in use.
Nothing, of course, is impossible. But if you wanted to make this work, you would have to do something like spoofing the CANbus via an Arduino or similar small computer. To do this, you would first have to understand the protocols that the radio is expecting, which are not documented. A science experiment, obviously.
Not really practical, as far as I can see.
Formerly: 2005 Airstream Interstate (Sprinter 2500 T1N)
Now!: 2014 Great West Vans Legend SE (Sprinter 3500 NCV3 I4)