hi kalaman,
You require a Vbattery detector circuit that will have a upper and lower voltage threshold level.
As you say the battery will discharge down to about 10.8V. [this is the lowest discharge voltage that is recommended].
If you used a PICs ADC, you could detect when the battery/load voltage is in the order of 10.8V and disconnect the battery from the load.
It would remain in this disconnect mode until you pressed a push button 'Reset', which would also be detected by the PIC and the load would be reconnected.
It would be also possible to detect a fully charged battery state to do the same reconnect action. If the battery voltage dropped quickly the PIC program could detect this condition and disconnect the load and await a manual 'reset'
By using a PIC it would be possible to build/program a 'smart' battery status monitor.
Does this give you any ideas?
Eric