Tesla's powerwall batteries ar made by Panasonic as far as I know.
I've I stalled my own solar array on my garage, 5kwh; enough to power my house year round, 3 small kids, but being in the Netherlands we use gas for heating, and no AC needed.
Having a system that can operate as if off grid, requires specific equipment.
Especially the converters, not all can run in island mode. Most conversters are made for domestic use, and use the 50 hz signal from the grid to generate their own frequency, basically 'fitting' their electrons nicely between the others.
(Which isn't how it works in physics, but for understanding it's a nice analogy)
If the grid quits, these converter quit as well. Second reason is safety; if the grid goes out, parts of a house would still be live, but you wouldn't know where/which.
So, Dang in your case ask specifically for a system that can operate in island mode. It would require dedicated wires to your ac setup.
But I would advice you to install it so that you have the option to switch from island to grid connected at your desire.
You're still making electricity when your AC is off, and the planet will thank you donating that to the grid. (Reminder, half of the dutch live under sea level). Pehaps you even get some income out of it.
If later it ever gets taxed, you just disconnect it, and benefit in summer from your investment and free, green electricity.
In NL the rule is that you dont have to pay taxes on purchase, but only if you register them. So every body registers, it saves 22% tax.
Benefit for the government is they know exactly how much KWh is installed where, and being able to upscale the grid's infrastructure accordingly, but also giving the possibility of future taxation based on capacity.