Yes, while charging your car, the home battery control will try to minimize the consumption of the grid by using energy from the home battery. However, since the car's battery is much larger than the home battery, the home battery will deplete quickly.

When the home battery runs out, all the energy needed is taken from the grid. It is not possible to turn off the home battery for the charging station because there is no direct communication between the two. Adjusting the home battery arrangement offers no benefit.

After all, your employer reimburses you for the energy consumed, regardless of the source (such as home battery or solar panels). Moreover, since your car is probably the biggest energy consumer during charging, it makes sense to use the energy stored in your home battery for that purpose.