Because you have Sunny Boy, Sunny Island is exactly what you want.
Alternative, Sunny Boy Storage.
Either of those will AC couple with your Sunny Boy and control its output to match loads and battery charging needs during grid outages. Changing a parameter in the inverter may be required to enable this function, but I think the new models recognize the battery inverter. Just need a data cable, possible add-on card for the interface.
For Sunny Island, either buy two SI6048 and connect for split-phase 120/240V,
or buy one Sunny Island and get a transformer. That could be a 6000VA isolation transformer, 120V to 120/240V, or a 3000VA 120/240V auto-transformer (two 120V windings connected in series.)
Each Sunny Island can pass up to 56A through its relays.
Connect:
Main panel <--> 2x SI6048 <--> PV breaker panel <--> "load shed" relay --> protected loads panel --> your circuits (refrigerator, computer, etc.)
Also, PV breaker panel <-- Sunny Boy
The "load shed relay" is a 48VDC coil, 2PST relay good for at least 50A, preferably 100A.
In the event battery gets run down to low voltage disconnect (70% DoD for my AGM batteries), loads are disconnected to prevent further discharge. Sunny Boy still gets AC so it can work when power comes up.
You can connect up to 4 Sunny Island, 2S2P, providing up to 23 kW continuous from battery, up to 44 kW surge to start motors, up to 112A 120/240V passed through from grid.
Supported 48V batteries are NiCD, FLA, VRLA, certain brands lithium with BMS that talks the right protocol. Some people here use DIY lithium; there is a BMS which works. From 100Ah up to 10,000 Ah, 5 kWh to 500 kWh.
Your other choice from SMA is Sunny Boy Storage.
It uses 400V lithium batteries of certain brands. I think about 4 kWh up to about 60 kWh are available.
An automatic transfer switch "Automatic Backup Unit" is required for off-grid backup operation. Otherwise, it is just an "AC battery", shifting time of use but shutting down during grid failure.