This is a normal situation in most boats (and probably RVs).
In my boat, I have a 450 Ah battery bank, a 2000 watt inveter, and a 100 Amp alternator on the engine. While underway, the alternator is charging the battery bank, while the inverter is drawing current from the battery bank to make 120 VAC to run my navigation computer (and any other 120V devices I may want to use) and also supplying power to various other 12 volt loads.
If the charger (alternator) can provide more current than the inverter and other loads require, the excess current goes to charge the battery bank.
If the loads require more current than the alternator can supply, the battery supplies the difference.
You can't really charge and discharge the battery simultaneously - although it may look like you are doing so.
As the charging source and loads vary, the battery will be charged or discharged as needed, changing modes automagically.