I may be missing something here since I have never seen the Heartland inverter installation. Usually there is a sub panel installed that would have just the circuits in it that are to be run by the inverter (refrigerator, TV, +). Again, usually, there is a circuit breaker in the main 120v electrical breaker panel that's labeled "inverter" (maybe a 30amp breaker). Again, usually, power from the pedestal goes to the main breaker panel, and across the 30amp inverter breaker directly to the inverter. From the out on the inverter, the cable runs to the sub-panel powering the "inverter circuits." On the inverter, there should also be a DC in from the battery. An inverter internal transfer switch, when it detects pedestal power, transfers the pedestal power through the inverter to the sub panel and also turns the inverter into a battery charger for the inverter battery. When you remove pedestal power, the internal transfer switch detects the loss of pedestal power and transfers the power souce to the battery then up to the sub panel.
The unknowns to this question are: do you have an electrical sub panel with a circuit labeled refrigerator? What kind of inverter is installed (charger or no charger). Inverter control panels typically "light up" when shore power is attached. The lights may indicate the state of the batteries, ie charged fully, currently charging, etc. Usually there is an invert button on the control panel. Usually, invert is not automatic after removing shore power; you have to push the invert button to turn it on. While many, if not all larger inverters have 110 outlets on them, few, if any have remote outlet wired directly into the inverter with romex. Usually the inverted circuits are in the sub panel connected by romex.
Pictures of your electrical panel and sub panel would be helpful as well as pictures of your inverter and inverter control panel. By now you may have already solved the problem. Good luck and let us know what you find.