Most inverters take about 1A on standby so it doesn't make much sense to have it running all the time. That equates to 24Ah on its own. If you buy a 12v fridge some of them (Alpicool for example) also come with a separate 240v psu in case you wish to power it from 240v directly. Mine is wired so that the fridge is powered directly from an EHU connection when available but uses the 12v system at other times. Using a pair of relays my fridge will automatically choose between EHU, alternator or habitation
battery. I can supply a wiring diagram for that if required.
Having run my Alpicool on a 13 day trip with no need for ehu I noticed that, although the quoted current demand is 8A, in practice it is drawing only about 4A most of the time when it is demanding power. The amount of time it is using power is less than half of my previous absorption fridge which took 15A. Reliability is also better. Failure of absorption fridges during gas operation is quite common due to corrosion blocking the jet.
Alpicool also provide an app so that you can monitor/control the fridge from your phone via Bluetooth.