Using a starter solenoid to operate a Q is a poor idea because of the way a Q is used. The usage pattern of a Q siren does not model that of an automobile starter at all. A starter relay is used once for just few seconds to start a vehicle, and usually has hours to cool off before it is called upon again to start the vehicle.
Think about your average response. Is it 5 minutes, 10 minutes, more? Is the Q wound up just once for the whole response. Probably not. It's probably wound up and let to coast at least a half a dozen times and probably more, depending on the time of day and the demographics of your response area. Just imagine how quickly your starter solenoid would go out if you had to use it to start the car at every intersection.
Everybody has heard someone who will "ride" the Q. This is especially bad for an intermittent duty solenoid like a starter solenoid. The solenoid never gets a break while someone is "riding" the Q, and can get very hot very quickly. Just imagine how quickly your starter solenoid would go out if you had to run the starter for your whole trip to the grocery store.
It is the nature of how a Q siren is used that indicates the need for a continuous duty solenoid.