Thank you so much for your detailed answer. I really appreciate it.Most HEV & BEV vehicles do have a standard 12V lead-acid battery to take care of vehicle "housekeeping" needs, in this case starting the petrol/gasoline engine. That 12V battery is ideally a "deep cycle" type. The gas engine is just like any other in that it requires significant cold cranking amps (CCA) to start the combustion cycle and thereby the standard 12V battery configuration. Typically the 12V battery is kept topped up by the vehicle's Energy Management System (EMS), drawing off the High Voltage (HV) battery as necessary, via DC-DC converter. There is also typically an Alternator as in any gas vehicle, but usually in a HEV/BEV it is electrically activated, although in some HEV the more typical belt driven off the gas engine when running.
The DC-DC converter however is more related to the HV battery system in that it accepts the high voltage of DC Fast Charger (DCFC) EV Supply Equipment (EVSE) and converts it for battery replenishment, etc. Be reminded that the HV charger itself is built into the vehicle and the EVSE is rather just the power supply conduit (btw: some home EVSE's are "smart" with App's etc., while others simpler).
I don't own a 250e so speaking generally from my own current BEV experience and some prior HEV.