There are several battery based inverters that can work in an AC coupled setup with an existing grid tie solar inverter(s). Only a few are Rule 21 compliant now, but if you have a setup where the battery is never going to back feed to the grid, it does not need to be Rule 21. While on grid, the existing solar can still back feed as it does now, and it is still legal. If you are only using the batteries for backup power, it becomes pretty easy. I am also trying to do time of use shifting, that is a bit more of a pain. Look at inverters from Outback, Aims, Schneider, Sol-Ark, Magnum Energy, and yes, even a Tesla Powerwall 2. I have a Schneider, and a co-worker of mine put in a Panasonic system. Others on this forum have the Outback Skybox, SMA Sunny Island, and Sol-Ark systems. I would certainly use a UL listed inverter. Pushing around a few thousand watts is not trivial. Use quality gear and install it to code.
How much power do you need while the grid is down? You need a reasonable estimate of your peak power in Watts, and your energy for a day in Kilo Watt Hours. From there we can help you figure out how big of an inverter you will need and how much battery.
Most Battery based inverters being used for AC coupling need to be as powerful as the solar you intend to connect at their output. There are some exceptions, but that is a good place to start. If you have a very large grid tie solar system, say 10,000 watts, you may want to split it and only have 5,000 watts when you go into battery mode. But if you have a small array like my 4,000 watts, you can easily back that up all of it with a battery based inverter. What it comes down to, is all of the solar power has to be able to go somewhere. When on grid, any power you don't use goes to the grid. When the grid is not there, it will go to your battery. With 4,000 watts coming from solar, if there are no loads running in the house, that could mean 83 amps of charge current to a 48 volt battery bank. That 10,000 watt array would mean 208 amps. The battery needs to be sized to take it.
How long do you need to run on battery alone? If you lose power, will you still have good sun? During the fires here, the smoke was knocking my solar production down to less than 1/2 of normal output. So plan on enough battery for 2 days, and enough solar to charge it all back in just 1 day.
With lead acid batteries, you can only use 1/2 of rated capacity or you will damage them and shorten their life, and they can only take 1/10 of their amp hour rating in charge current. A 100 AH battery charges at 10 amps etc. Lithium batteries can use most of their capacity safely, and can also take a lot more charge current. They do cost more, but with a proper setup can last much longer, so the cost may break even or possibly end up cheaper in the long run.