Most alarm systems works on a Fail Safe configuration, i.e. the system is always powered up. In that way when someone cuts any of the wires, the system sounds an alarm.
If you only power ON the system (i.e your transmitter part) when opening a door, you will have no monitoring of the wires, and if for some reason the wires are cut or broken, you will never know, and the alarm system will not operate as desired.
I understand your need for saving battery power, but you'll have to find a way to make it fail safe.