Most of the commenters so far seem to think that the line you labeled "12V" is a power supply. However, I take it that you mean that it is a control signal that is pulled up to 12V — presumably through a pullup resistor of 2400 Ω, since you also specify a current of 5 mA.
Your second circuit is correct, and 110 mV is perfectly reasonable for a saturated NPN transistor @ 5 mA. If your equipment really requires a lower voltage drop than that, then you'll have to try either a logic-level MOSFET or a mechanical relay. Are you saying that the radio is failing to mute using the transistor?