Apparent power for a grid inverter

I have an Huawei SUN2000 3-phase Grid inverter. The output current for every phase is nearly the same, regarding the load on each phase.

If I would buy a Luna2000 Battery, would the output on every phase asynchronous and would it match the consumption of the load on each phase?

How would the inverter handle inductive load, like in the night when there is no PV output and the fridges turns on? Would the inverter output apparent power on one phase, or would there be a synchronous output on every phase without reactive power?

In Austria we only got charged for real power and not for apparent power, so we would waste money if the inverter have to output apparent power.