Improve a simple “magnetic alarm” with a timer, to extend the ringing time
I assume the batteries and the 12V have common ground. If that is the case:

I would do it with a pic10f200 microcontroller and an nmosfet. 

The nmosfet would be controlled by the mcu and would replace the reed relay. The relay would be connected to the mcu directly with a pulldown resistor.

The firmware can delay the switching of the fet freely.

The mcu can work directly from the 4.5V.

The pic and the fet would cost ~$1 together.