i have run my microwave off my inverter many times until the low voltage alarm has gone off on the inverter.
when i shut the inverter off the voltage on the battery recovers and i can run my lights and tv for hours after that before the battery is truly discharged i have a automatic 10a battery charger that float charges my 2 12v house batteries any time i am not using this camper van and they also charge as i drive it.
i easily get up to 7 years out of a set of batteries so just because the alarm goes off on the inverter under load does not mean the batteries are totally discharged.