You haven't told us the voltage. That makes a difference.
2 awg with 90 degree C insulation is good for 190A if single conductors in free air.
We would normally use heavier gauge wire for the inverter, unless it was at least 24V.
For 2 awg I would use 200 A max, because that is rounded up from 190A.
I normally use 125A for 2 awg when it is in a cable.
As for what breaker and wire should be used for the inverter, example calculations are given above.
Except, I've figured out that the current draw from the battery has a massive ripple, so its RMS value (and heating of the fuse or breaker) is more than those calculations say. So I would want it to be even larger.