Many inverter systems, and all of the higher end units have an adjustable low voltage cut off setting, so no need to kill your batteries. In my case I set that pretty low, as the sole function of that battery bank was the inverter.
I very much like having the inverter ON in case of a power outage, to keep the refrigerator in particular going. On our set up, we could get fancy if we wanted and have the generator auto start at another predetermined battery voltage (and on newer systems that can be triggered by state of charge as well) and then shut off after some designated period. Don't think we've ever used that on shore power though, it's a bit complicated in that case vs being out on the hook.