L;
I continue to suspect the wiring between the battery and inverter to be too small. This causes too much Voltage drop and causes the inverter to alarm during engine idle. Use a Voltmeter at the inverter terminals to check.
A 580W cooker will draw around 48A. By comparison, how many Amps can you draw if you turn on every electrical item(brights, fan, AC, defroster, etc) at the same time?
The Kill-A-Watt did accumulate the Watt-Hours automatically as KWH. The reading is selected by the far-right button which toggles between KWh and Hour.
Recharging a deep cycle will take longer than a car battery. Your driving time might be a problem. Dual purpose would be a better.
Solar will charge the battery(any type) whether driving or parked.
Will you consider a solar system?
HTH;
John