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The Hall Effect sensor referenced in this forum thread would work for you, and it does not need a relay. You can run it on any voltage above 3.8V; say a 9V battery. You need a current-limiting resistor in series with the led.
The LED bit is just to show a rough estimate of the amount of power
that will need to be turned on and off. I don't know the exact amount
and I don't want to have to figure it out - I do know that a relay will do
the job though.
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.
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