Upon further testing, we're both right, sort of.
When the scanner is on battery power, the 3.7V pad is switched. If the scanner is off, the 3.7V pad is off.
But when a powered USB cable is plugged into the scanner, the 3.7V pad is constantly on, regardless of whether the scanner is on, off, or off and charging the batteries.
Running the GPS module off the 3.7V pad will not drain your batteries when the scanner is off. When there is USB power available, the GPS will be active. This will keep the GPS module's backup battery topped off, which will accelerate getting a fix when you start the scanner on AA battery power and the GPS has been powered off.