Your Inverter is wired directly to the battery so putting the Use / Store switch into Store will not power off the Inverter. The Inverter fan is temperature controled usually so that means the Inverter is getting hot.... either from use, not enough air for cooling where it is installed and / or very warm air temperatures.
You need to turn your Inverter off at the control panel for it to truly be off. Now if your inverter has a built-in ATS, it can still be powered on when plugged into shore power even if the control panel for the Inverter is in the off position.
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