The battery will run a fridge but not an LED bulb?
The inverter can produce around 3.5 kW and the house can use well over that figure, so one can't have the whole consumer unit connected to the UPS supply, the items connected need to never even without a power cut exceed the 3.5 kW total. So the essential items only have been connected, central heating, and freezers.
There is a spare socket upstairs and nothing stopping me from plugging in a standard lamp, however we also have rechargeable torches which auto turn on with a power cut, and to remove lights from main consumer unit and fit on the UPS means we would have items like an 8 bulb chandelier just for living room lights which is some 48 watt just for the living room.
Power cuts are not that common, and the main problem is one loss of food due to freezers failing and loss of heat due to central heating failing, we could light the open fire, but we have it loosely boarded up so in the summer the portable AC exhausts into the chimney so it would not be easy to quickly remove it and light a fire in the grate, and we don't have that much wood to burn.
So while we have power the battery can supply whole of the house, and by midnight it is normally turning off as down to 5% so at the moment the solar starts around 8 am which is only at the moment 22 watt, by 10 am we are getting a descent amount, and by midday we can be getting 5 kW, if we have a power cut we would want to reserve that for essentials, but other wise used for all, the only items when we always import is the showers, all other items are covered with solar.
However we do get dull days where hardly any solar is produced, if we were unlucky enough to loose power on one of those days then we would still loose freezers and central heating, but the chances of going away and returning to find freezers all defrosted is very slim now.