It could be a matter of wire size which contributes to the above mentioned voltage drop.
I suspect that as the radio transmits, it is "robbing" just enough juice and the controller for the siren/lights is sensing that and is shutting itself off.
There are many variables with your set-up. The type of siren (mechanical/electronic), the type of lights (LED, strobe or incadescent) along with the power of the radio.
I'd suggest running a separate circuit for the radio, separate circuit for the lights and separate circuit for the siren. The size of the wire for the circuit should be consistent with the power requirements as stated by the manufacturer of the radio/lights/siren. (There are charts on the internet where you can find which size of wire to use based upon the current draw).
Glass type fuses are really old school when it comes to emergency vehicle equipment. Nowadays either automotive type circuit breakers or ATC fuses are used for over-current protection.
Lastly, just to cover additional possibilities (and they could be remote). What is the condition of the car battery (does it have enough stuff to handle the emergency equipment)? Also, how about the alternator for the vehicle - is it functioning properly and putting out enough stuff?
Be sure that all your emergency equipment has good grounding. While chassis grounding will suffice typically (there are those who suggest running ground wires back to the battery itself), connections can be come loose or get dirty with time.