The charger part of the inverter charges the batteries.
I believe they switch all AC loads from inverter to grid when plugged into the grid, and the charger then 3 stage charges the batteries, automatically.
I think some of them are very good chargers with programmable setpoints.
i cannot say I am familiar with them and exactly how they function.
I am more of the separate inverter separate charger mindset as I don't really want anything mindlessly automatic
Seems the inverter chargers are $,$$$