If I got your assumption right, you are counting 100 kWh for 400 km.
In a country like France, with something like 30M cars doing 12,000km per year (ball park figures, but not too far off), you need 100 billion kWh, i.e. 100 TWh for the whole fleet.

French electric demand right now is around 500 TWh. So we're talking an increase by 20% only, a lot of it which can take place at night, as you mentioned. But a smoothing out of demand will actually make the integration of renewables easier, as the capacity to deal with intra-day changes will still be there and will be just as easily able to deal with intermittent renewable generation - it's just that gas-fired plants may work at night, to replace solar, than during the day, as a top up to nukes...

Wind power