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When you are plugged into shore power, your AC system should not be operating off the inverter. It should be being fed live from the shore power. The only time you should be using the inverter is when you are NOT plugged into shore power. If your boat is set-up so that you have to use the inverter to run the stove/fridge even when you are plugged into shore power, then that is highly inefficient and you should have it corrected.

You should have a panel for both your DC and AC loads. The AC panel should have a large master breaker for activating shore power (the DC panel's "master" switch is usually a separate battery selector switch). Normally the AC panel will have a dedicated breaker to turn on the battery charger as well. Properly installed, the battery charger should charge all battery banks (house/domestic and engine/starter). The AC panel woudl normally also have a breaker that turns on power to AC outlets.

If you are using the inverter at anchor, be sure to turn it off before anyone goes in the water swimming from or around the boat. And shut off all AC at the master panel.

EDIT: The above comments are generic, not specific to your model/brand of boat. Also, when I use "AC" above I mean alternating current, NOT air conditioning.