It would help at noon, when usually inverters make so much power they push the local grid to saturation which causes them to disconnect from the grid for a while. There are different rules for this for different localities, typically the installer will pick the regime appropriate for the local grid. Where I live (NL) the voltage is allowed to hit 253V for up to a minute, if it takes longer than that my inverters will switch off. That's a requirement to pass inspection, if your inverter can not do this then that's an automatic fail.

There are 'home batteries' that store power for practical reasons up to a day, maybe two. They are useful for sunny/cloudy conditions to maximize the amount of power locally consumed. But battery storage and retrieval isn't free and so far - at least, for my installation - this does not make economic sense. But another factor of two in price drop for my preferred chemistry and it definitely would make sense and then I would probably install about 20KWh worth of capacity. The trick then becomes to balance power sent to the grid and sent to the battery so that you maximize the utility of the grid in order not to wear out the battery prematurely. That means cycling the battery between 70 and 90% state-of-charge for Lithium-Ion. I'd much rather have Lithium Titanate so that is what I'll be holding out for, they are already available, but still too expensive. But they're much safer and have far better charge/discharge curves and life-span.