Sometimes, depending on
duty cycles and
load staggering, you can pull quite a few more amps through wire than the chart says. Alternative energy systems should be thought of in a different way altogether than grid energy. With grid power, the electrician will usually wire the house so every circuit can pull it's full amperage continuously. This is because there is always enough energy available, it's relatively cheap in the short term, and most people are simply too uninformed/uneducated about power consumption to use it wisely. I've seen a retired electrician run 200 ft of 14 gauge extension cord to his green house to run two 1500 watt space heaters, a fan, lights, and a water mister. It's amazing that his house didn't burn down. He wanted to put new ends on the cords because the old ones melted......
My system uses three runs of 2 gauge wire from the batteries to the inverter. It handles 300 amps continuous with no temp rise at the wire and the only about five degrees over room temp at the 350A breaker. I pulled 450A through it with the 5kw inverter and the wire got slightly warm to the touch....didn't have the infrared thermometer yet. The wire chart says I should have used bigger wire but I had to use what was available and what I could afford, thinking I'd monitor it and upgrade if necessary. I also considered that the only time my inverter will be pulling hard on the cables will be when I'm running things that are very intermittent, not giving the battery cables time to heat up. The three runs of 2 gauge works perfectly. Maybe the wire has better conductivity than wire that the universal charts reference??? Maybe everything's drastically underspec'd these days because too many people haul pallets of bricks on top of their VW Jettas? I don't know but I trust common sense, experience and real world testing more than charts most of the time. If I know I only have a 150 amp breaker, or wire with a 150 amp capacity, I won't plug in ten devices all at once that need 20 amps each, but I might put an inverter that can pull 200A on that wire if I know I will only be using it at that rate intermittently.
So what I'm getting at is, your family doesn't have to run everything at once. The toaster(is toast really that important...must be good bread..), the washing machine(once a week at most, don't get so dirty!), the dishwasher(I can do a load of dishes with one gallon of water in five minutes...dishwashers

....), the furnace, ten LCD tv's, the stereos, computers, every light in the house......everything being on at once shouldn't ever happen. There's no reason for it unless 50 people live in your house. I've seen many homes running like that though. Most likely, during the day, people are working. The panels are charging the batteries and there shouldn't be much load on the system. At night, a few rooms need lit up, some water needs heated, the heater blower might need to come on ever so often. Everything after that is pretty much an excess that can be moderated. If you're on the computer, turn off the TV, etc.... Big energy hogging chores can be done on days when the battery bank is pretty charged up and there's good sun.
It's actually really nice to know where your power comes from, how much you have, and how fast you are using it. Once you are aware of those things, you can make intelligent decisions and cut your energy usage by 90%, making alternative, sustainable power sources much less expensive to buy, install, and maintain.
Toast.....