Without trying to answer your "I believe the setup circumvents the fee" implied question, which you should find an actual answer to...

See reddit's r/solarDIY. There are a lot of setups like this, either for full house power or partial. I have a relatively big server in my basemeent that I power this way, for example, with a battery system that acts as both UPS and a "charge the battery from solar and run off inverter when possible". Something like an off-grid EG4 is popular among the reddit crew as a way to manage this; the device tries to run locally but accepts a shore power feed from the grid so if you need it you draw grid power. No grid tie involved.

> I believe that setup circumvents the fee since the only thing using energy is the battery, as though it were an off grid setup.

Entirely based on your jurisdiction, but that kind of zero-export requires the same permits and connection fee as a full net metering in mine. You can't end run the process by just being sneaky, electrical work needs permits and they'll either make you take it out (!!) or bring it into full compliance.

As an example, my utility is one of those ones that requires a separate manual disconnect, even though the vast majority of all-in-one inverters have a built in disconnect. It doesn't make any sense and it's at least two grand in permit + equipment, but that's the rule.

Having just put a battery into my existing system. For the power company I use the answer is no to your question. Many power companies charge that fee depending on the size of panels. Now if you do not hook to their system they do not care (off grid). But if you plug into it then they care and charge for the privilege.

Now you could pull particular items out of their grid. Just need the wires and panels to make sense. They would not care about that either. It is only if the panels are plugged into the grid they start to care. Downside is sometimes waiting for that battery to have enough juice to work. It goes faster than you think and some battery systems are decently expensive just by themselves, on top of the panels/inverters/controllers/install.

Ask a lawyer about the law.

That depends on what options are available to you for residential solar installation. Installing solar panels + micro-inverters is very cheap in cost of parts, but possibly expensive in labor. It will easily pay off your grid interconnect fee every month once you hit break-even from operating for a few years. Financing vs cash, cost of installation, tax credits, are the biggest factors in how quickly it becomes profitable. If you are extremely handy you can DIY most of it as a homeowner (depending on local/state laws) which cuts costs significantly.

You can do a whole home backup + solar panel input (aka ESS+PV) but it's much more expensive in parts and installation and is more complex. If you don't add grid interconnect it can be much cheaper as you can use "off-grid" parts (grid interconnect requires more expensive parts certified for certain standards).

But some jurisdictions may not allow cheaper "off-grid" parts/systems to be connected to the grid, even downstream. So an alternative would be to install a transfer switch, where you manually switch from the grid to off-grid. But then you literally have to choose whether you use the ESS+PV or the grid at any given time and manually throw a switch (you can't use grid to charge your system). And yes you can set this up for only specific circuits/appliances. Again it depends on your state and local laws.

AI can do most of the research for you. Keep a Git repo and add documentation and details to it over time, and you can build up an analysis of what works for you based on all available options, price comparisons, energy requirements, and local/state laws. It is unfortunately quite complicated because there are many options but not all will apply to your location. And before you do any of this, do a solar site survey to see if you can even reliably generate enough power from the sun!