.......I often stay on sites with only a tap and a toilet emptying point and I use inverters in the way the OP suggests to produce mains power.
However you have to remember they do not produce electrical power.....they just convert it for use in an appliance.
It is pointless having a 3 kilowatt inverter unless you have a means of producing 3 kilowatts in the first place.
Such an inverter would need a massive battery bank to be of any use if you don't.
I use a small 150 watt modified sine wave inverter that gets it's 12 volt power from the caravan leisure battery to power a mains TV and Sky box and any small mains charger such as for a tablet or phone.
Don't use an inverter to charge an electric toothbrush.......in my experience it will destroy the charger.
I also use a 600watt pure sine wave inverter which can be temporarily connected to the tow car battery to provide mains power to the caravan via the normal EHU socket using a suitable connecting lead.
The 12 volt leads from the battery to the inverter have to be short meaning the inverter has to be next to the battery. The mains wire from the inverter to the caravan can be of any suitable length.
This set up is used to charge the caravan battery via the on board smart charger or to power a 500 watt hair drier. It will also power the fridge but I never make use of this.
The tow car battery is monitored and never allowed to fall below 12 volts to make sure there is power left to start the engine.
So long as the tow car engine is run on trips out from the camp site fairly regularly to recharge the battery and gas is used for cooking, heating, hot water and the fridge, then I can survive without EHU indefinitely winter or summer.
🙂