Your last answer suggests you may need to research pass through more clearly? Circuits supplied by an inverter with built in pass through are automatically supplied by shore power once it is plugged in, no battery cycles involved unless the shore power goes out & you have the inverter function enabled. The inverter automatically switches the hot & neutral to shore power- that’s the pass through. This means (if you have pass through) plugging your microwave & fridge into the inverter outlet should automatically be running off shore power when plugged in at the dock. If the inverter function is turned on, they will run off the batteries via the inverter. If you have a generator, they will run off the generator, not batteries. (Your shore power/ generator switch will switch both the hot leg(s) and the neutral leg to the source you select).
The issue with the neutrals/isolation transformer is related but separate. Typically you won’t know of neutral commingling until you plug into a new marina or recently rewired marina with the new, more sensitive ELCI’s. These devices monitor hot & neutral current & trip if >30mA discrepancy.
Your inverter with pass through, switches both the hot & neutrals to the shore power. But most boats have some circuits that don’t run through the inverter. The rule is that the neutrals must be grounded at source. On shore power, that’s at the main dock breaker. On the boat while not plugged in, that will be at the generator or the dc ground. This requires that neutrals from inverter circuits must connect ONLY to a separate buss/inverter neutral. Otherwise at the dock, plugged in, neutral commingling between the inverter & non inverter circuits allow some current to return to shore via an alternate path which triggers the new & more sensitive ELCI’s. It is very time consuming but relatively straightforward to check each neutral is returning to the correct buss. Other culprits include the neutrals for the AC panel LED lights and some appliances meant for shore use that have neutrals grounded to their chassis. AC outlets are easy to test. Hardwired devices will require more work.
Because this is a time consuming process, many just elect to install an isolation transformer instead.