Like Wheels points out many inverters "drop out" when detecting 10 Volts at their internals. Most good quality low frequency inverters (they switch at 120 Hz US 100 Hz. Euro) have a 3X power surge capability and, therefore, if you have loads which push that limit for starting or peak loading you should size the cable accordingly to prevent excessive voltage drop to the inverter
Keep in mind that the voltage sag at the inverter is due to cable voltage drop as well as the voltage drop across the internal battery
resistance, which is often substantial and overlooked.