Instead of using a generator as backup for consecutive cloudy or rainy days, can I just use household mains instead? Ideally it would be nice to just somehow plug my inverter into an outlet permanently, and have the mains power automatically take over if batteries fall below a certain charge (e.g. 30%). Does such a connection exist?
Yes, that feature is already on the inverter you chose. You simply need to hook up the wires according to the instructions.
Consult with the manufacturer whether the sequencing will work the way you want, and that it is compatible with lithium batteries, and will know what 30% is on a lithium battery.
I'm honestly not a big fan of this unit, since it has an RCD and many modes that require manual intervention to clear. It does not seem fit to run unattended for long periods. You will need refrigerator alarms.
I do not agree with the idea of "all in one" combo devices that combine the function of charge controller, battery charger, inverter, transfer switch, And every other darn thing. They are more difficult to find, they tend to do everything in a mediocre manner so your choice is limited. It takes ages to find the perfect unit that does all the things the way you want them done. Suppliers are very limited, takes weeks to arrive... and then later, it breaks. Now you are entirely down, because you put all your eggs in one basket. It's more difficult to troubleshoot with so many functions rolled into one. You find the unit is discontinued, and you must start the product search all over again while you're down. It takes weeks to arrive... and then it fails suddenly.
I recommend a discrete charge controller (perhaps two, one per solar panel bank), a separate battery charger fed from AC mains (the better way to do a "transfer switch"), separate inverter. Now if one goes down, only that one has down - if you lose the charge controller you can still charge batteries off mains. Seeking a replacment is simple and wide variety of products will do.