It all depends of the fridge. If someone has fitted a domestic fridge then it will have been designed to run from the mains and no consideration will have been given to running it of 12v. You need to check the specification of the fridge in question.
Of course you can - but you need to leave the domestic battery bank on which means if the shore power trips while you are out and about you could end up with a dead battery. If the fridge is connected via the inverter the worse you get is rotten food ;-)
I'm assuming here that you have a setup similar to mine which comprises a mains battery charger and a 12 - 24v DC fridge that can run directly from the domestic battery bank but is also wired to a 240v ac - 24 dc inverter