Cut off of discharge?
Most convenient would be if inverter monitors SoC and has setting to shut down.
If inverter has remote enable, wire to a monitor, possibly SCC if it has SoC controlled relay.
There are breakers with remote trip (Midnight has some), but probably not high enough amperage for inverter battery current.
Could disconnect AC loads with relay or remote trip breaker. Then still have inverter idle load. If it has standby/search setting, that would further reduce draw.
There are relays for battery current (Gigavac, etc), such as used with BMS of lithium battery. You want low hold current, and "economizer". And it has to be rated for continuous operation, not like an automotive starter relay.
You can control relay with a battery monitor, maybe just battery voltage but it needs a delay to allow surges. And voltage will give poorer measure of SoC, depends on both current and temperature.
Controlling loads so heavy loads are only operated with higher SoC would help avoid draining battery to cutoff. Or just enabling them when there is sunlight, by sensor or timer.