Solarpunk

Balcony solar does send excess power back to the grid. If you have a really old electricity meter it might count backwards when that happens and you might be legally required to get it replaced before installing the panels (depending on local laws).

A battery that sends power directly to your entire household but never back to the larger grid would require integration with your electricity meter (not just plugging it into a random outlet) because you can't measure the electricity usage in your home from a random outlet and know when to discharge.

Have you considered the energy loss. Depending on your battery you might end up with like 75% of the energy you paid for.

Because both storing energy in a battery and taking that energy out is not 100% efficient.

I think ecoflow has options to charge from the grid and discharge later. It's called time of use rates. I know it's on their regular power stations and the smart electrical panel but not 100% certain it's on their balcony solar line called 'stream'. They also have manual and automatic settings to charge up from the grid before large storms, that you might be able to hack to accomplish your goal

They sell larger batteries for camping (and power outages etc.). I didn't think it'll do the specific charging your looking for but, that all sounds doable with some extra hardware to regulate the charging.

I am not aware of a cheap stand-alone unit that does this, but the typical solar hybrid systems can be used without solar and configured like that.

Uninterruptible power supply is what you want. It basically trickle charges a large battery and swaps to the battery when you lose power. They can last for a few hours depending on the size of the battery and the load on the circuit.