I fixed a screw-up in the diagram. Look at it again.
The sensor will sink 20mA. That is not enough for a relay coil, but it is enough to drive the base of a switching transistor. You will need a PNP transistor to drive the relay coil. Forget using a small battery. The relay coil will require enough current to where you should use a power supply like a wall wart. You need to recalc the LED's current-limiting resistor for 12V. You dont need a watch battery, just put the led/resistor in parallel with the relay coil.