Re: Battery to Inverter Wire Size
What matters is the current, not the battery bank size. 2/0 can handle up to 275 Amps (you will see different ratings depending on application).
How big is the inverter? Roughly 2kW judging by the recommended fuse size. That will draw about 167 Amps under most circumstances, up to 200 if the Voltage is allowed to go down to 10, and momentary surge above 200. Why so many different current numbers? Because of the variable input Voltage, adjustments in Ampacity rating for application, NEC fuse size calculation, et cetera.
So rather than try to learn all this stuff in one go (like trying to swallow the whole bottle of aspirin at once) accept that the people who made the inverter already know it. If they recommend a 250 Amp fuse, use one. If they recommend a minimum 0 AWG wire, go with that or larger. Don't go with a larger fuse size and don't expect to see 200 Amps coursing through the wires either.
1220 Watts of PV, OB MX60, 232 Amp hrs, OB 3524, Honda eu2000.
Ohm's Law: Amps = Volts / Ohms
Power Formula: Watts = Volts * Amps