Yeah, relays have a ton of uses. I've done sooo many cool, crazy, and useful things with them. Well... your idea is right on track, but it's not going to give you the desired effect... and here is why: The DEI voice module (516 L) requires trigger inputs and sometimes even a siren input. So it is tied into your door trigger, trunk, hood, etc. Heres basically how it works. When you arm your alarm, the ground when armed wire tells the voice module to turn on and start working. Whenever the voice module sees a trigger signal (usually a negative pulse) it then alerts you with a message. In a way, it's an alarm itself.
So lets play out your scenario. You hook up your relay as you have intended and when you turn off your car the relay sends out a ground signal to the voice module and powers it on. Well, when you walk back to the car to get in it, the second you open your door, the voice module is going to see that ground signal from your door trigger wire and it's going to activate the alert message. The same will happen if you open your trunk. What you need is a remote that controls that ground when armed signal so that you activate the voice module when you arm your alarm, and deactivate it when you disarm your alarm.
So you say you only have the factory alarm. Well I am 99% sure that we can use the clown nose wires to give us a ground when armed wire. Your clown nose works right now correct? Well what I need you to do is grab a multimeter and tell me what kind of activation the clown nose uses. What side of the led circuit is constant and which one is pulsed? How long are the pulses and what voltage. Once we get all that info, we are going to use this little device called a "universal trigger output module". It's made by PAC and called a TR-7. We use that in conjunction with a relay, and presto, we have a ground when armed signal. Once we have that, we can wire up your voice module all tied into your factory alarm.