Hi from a looong time lurker.
Ben the Prius uses a dc-dc converter to power all the 12V stuff whenever the car is in ready mode. It should not worry the 12V battery to run in EV mode for extended periods because it uses the hybrid battery to power the 12V (just like any EV).
I had a 2001 Prius and experienced times when the car would not go into stage 4 sometimes (driving on freeway shortly after startup) and would run the ICE even when in stop start traffic. To get to stage 4 in those circumstances you need to come to a full stop for about 10seconds. Sounds like you need an EV button?
Congratulations on getting a PHEV (even though its the wifes)