Yes, Lithium lead replacement batteries typically have a BMS (and at the risk of getting off-track, not all BMSs are equal) and are typically without inverter/charger comms.
My point is that the BMS is (typically) in the best position to make key decisions, and not the inverter. This is so even in cases where the batteries do have inverter / charger comms. An example of this is the Pylontech US3000 which issues a mandatory charge command to the inverter when the battery SoC gets dangerously low.
Inverters can like to be clever, but battery BMSs are more clever.