Installing a GPS module

If you connect it to 12V battery, it will drain it. Though the 12V on the OBD connector is switched, so when the Prius is off, it has to work on it’s own battery.

 

When car is off, Total parasitic drain more than 0.07 ampere is no bueno.

 

Connect it to the car's 12 V supply at a point which goes OFF when the car does.

The wires going to that 12 V (cigarette lighter) accessory plug that you don't seem to want to use might be a good place.

AND you need to be sure that the device actually accepts raw 12 V input.
Most have a tiny power converter built in to the supplied 12 V cord.