Charge controlers have an adjustable high voltage set point, so the batteries are not over charged. If you place a relay circuit that disconnects the batteries from the inverter so the voltage does not fall below the 12.06 50% level it would reduce damage to the batteries. The problem is that the circuit will need to not enguage the inverter until the voltage is some what higher like 12.30 volts. If needed the relay could open the 115 volt side and the inverter would only draw a small amount of current, until you checked the battery voltage level.
Thanks for any help in this matter.
Dale La Roy Smith II