What is the wiring connected to for power?
I power things through either a cigarette lighter or "power point" outlet in my 2008 F150 truck and they seem to have constant power. However, I've found newer vehicles (rentals and others including my wife's 2013 Ford Edge) to have more "safety" features that cause things that drain the battery to automatically shut off after a while - including USB ports and outlets.
You'll probably need to find a source that is directly tied to the battery (or tie directly to the battery yourself). However, this may not solve all of your problems either. If the voltage drops (i.e. when starting the vehicle), the radio is still likely to reboot.