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Normally, the battery disconnect is intended to disconnect the battery to prevent unwanted drain when the coach is in storage. The green LED indicates 120V power. If the switch is off, the 120V and converter continue to power the 12V needs. It will NOT charge the battery. If the switch is on, the converter WILL charge the battery and the converter and battery work together to supply 12 V needs.
If there is no 120V power, the green LED is off. If the battery disconnect is ON, then the battery will supply 12 V needs including lights and fridge control panel power. If the battery disconnect is off, there is no battery power to lights or refrigerator.
With this in mind, do your lights in the coach work without shore power? This tells us if the batteries are supplying power to the coach. This will help us to better troubleshoot your issue.
As a separate discussion, the genny may not be able to handle A/C and HW heater and Fridge all at once, thus bogging down. This is a 120 V issue unrelated to batteries.