Why does large 12v battery deplete quicker than 3 small AAA batteries
Let's see. Assuming the manufacturer isn't lying, the 12V battery has 7Ah, that makes 84Wh, which would translate to 16.8Ah at the 5V end at 100% efficiency.
Let's be generous and assume the car charger has an 80% efficiency (they usually aren't very good with low currents); that leaves 13.4Ah.
If the LED pulls 10mA, it would, on its own, drain the battery in 13.4 / 0.01 = 1344 hours or 56 days. This does not include self-discharge of the battery, the power consumption of the car charger itself, or the alarm's power consumption.
I would remove the LED and try again. Also, a more efficient DC/DC converter (which is the main bit in the car charger) would make a big difference.