A couple notes on your schematic.
First, you are off to a good start, there are a lot of elements I like in your design (for instance I like that most of your loads are DC native, this is more efficient with a battery based system).
Second, you asked about a fuse or breaker for the inverter, either acceptable, a fuse is in some ways better suited, a breaker is more convenient, either can work. What you definitely want to avoid is a cheap no name high current circuit breaker from amazon etc, best case they nuisance trip well below there rating, worst case they fail in a way that does not protect your system. Quality breakers from reputable brands generally top out at 200A unless you want to pay upwards of $100 dollars. Fuses can go much higher. If you use a fuse, a Class T, MRBF, or ANL fuse is well suited for main battery protection and/or the inverter circuit.
Four rules of thumb to keep in mind when thinking about fuses.
(1) the primary purpose of the fuse is to protect the wire
(2) any time you step down in wire gauge you need to step down in fuse size to protect the smaller gauge wire
(3) a fuse should be sized > 125% of max current of that circuit and < the current carrying capacity of the wire it protects.
(4) a fuse should be as close to the power source (beginning of the circuit) as possible
Third, regarding the shunt
A shunt based battery cannot accurately measure state of charge the way you have it wired. Think of current like the flow of water and think of a shunt as a device that measures how much water is in the water tank based on how many gallons flow in and out of the tank. To measure it accurately, all the water must flow through the measuring device. The same is true of a shunt and the flow of electricity. A shunt has to be positioned so all current flowing into and out of the battery is measured if you want it to accurately measure state of charge
How I would accomplish this by moving all the positive and negative connections you currently have going straight to your battery bank and instead using busbars. Then run single wires between your battery bank and the positive and negative busbar. The shunt would be between the negative busbar and the battery negative. This kills a few birds with one stone, it fixes the shunt measurement issue, allows you to use one main battery catastrophic fuse and optionally one disconnect switch, it is also a best practice not to stack a bunch of connections on the same terminal. Here is an example of what I am describing here (the blue square is the battery, ignore the details, it is a lithium diy battery so the details dont apply to your situation):