11.8v shut off is ridiculous. Most straight inverters alarm at 11v but keep running until 10.5v shut off. You have an inverter/charger which might have settings so perhaps you can change that as suggested above.
If you have a 1v voltage drop with MW on, now you need 12.8 battery voltage to get anything. How when a full battery is 12.7?
EG I have a 1v drop with MW on, but my inverter alarms at 11v so I can run it when battery is below 50% SOC at 12.1v. Your wiring is fatter than mine and we both have four 6s.
Edit: do you have some other thing in the circuit that shuts off at 11.8? Not sure how they work, but some monitors like Victron have a way to do that?