You might want to consider the following combined switch and low voltage cutoff
simulate this circuit – Schematic created using CircuitLab
When 'off', it consumes no power.
M1 and IC1 form a self-latching 'thyristor'. The TLV431 is a precision shunt regulator, that can also be used in other ways. Here, it is being used as if it was an NPN transistor with a precise 1.24v base threshold voltage and a very high current gain. When the output voltage falls such the IC1's input is less than 1.24v, the latch trips out. Set R4 to adjust the trip-out voltage, or leave out R4 and calculate R2 and R3 exactly.
You can use either the on/off switches, or the on/off toggle, or open collector logic to turn the power on and off. If you connect a capacitor across the on switch, you can make the circuit default to powered 'on' when it's first connected.
Beware that the TLV431 is an unusual device in that different voltage ranges are available in the same part number. The TI device only goes to 6v. You need the On Semi, Diodes Inc or the Zetec devices for a 16v version.