Backup battery switch with MOSFET?
If I got the requirement correctly then this can work.







simulate this circuit &ndash; Schematic created using CircuitLab

The mosfet is an enhancement mode P type.  

While the main battery is >13v LiPo saver outputs that voltage to the gate keeping Vgs close to 0 and the mosfet off.  

When the main battery voltage drops below 13v LiPo saver turns off (output floating) and the gate is pulled down from resistor R2 (Vgs of about -15v), the mosfet turns on and the relay switches to the backup battery.

Note that normaly the max Vgs is about 20v so 15v is a high as you want to go, if you expect Vgs to go higher (if the battery can be more than 15v) then you should use a voltage limiter (like a zener).