The 9 kW estimate was reached by assuming a discharge time of only one hour for the three fully charged good batteries.
When when a 12 volt battery, rated at 250 amp hour, is discharged in one hour, the load needs to be rated at 3 kW.
But if the battery is part of a daily cycle power system, to be discharged over 8 hours, the load will be only 375 watt.
If the three 12V batteries are wired in parallel, it will give 12V, with 750 amp hour capacity.
12V, 750 amp hours, divided by 8 hours = 1125 watt.
If the three 12V batteries are wired in series, it will give 36V, with 250 amp hour capacity.
36V, 250 amp hours, divided by 8 hours = 1125 watt.
So we are not necessarily discussing a 9 kW inverter here, we are probably discussing a 1 kW inverter powered from a low voltage. An uninterruptible power supply, (UPS), will usually convert a low voltage, (12V, 24V, 36V or 48V), battery supply to 115VAC or 230VAC.
Unless there is a good reason for discharging the batteries in 1 hour, a UPS is the way to go here.