I still think 4/0 is goofy for a 2000w inverter. I use the divide by 10 rule for DC amps so that would be 200 amps draw at full load.
By comparison, my Vector MSW 2000w inverter that is heavy duty for running motors too (now in the attic), has these specs for 6 ft and 10 ft:
1500w version- #2AWG, 2/0 and a 300w ANL fuse (they cover for the surge rating of 3000w it seems)
2000w version- 1/0, 3/0, and a 500w ANL fuse.
If you don't cover for the very brief surge time, you can use the ampacity/gauge for 150 or 200 amps.
On voltage drop, first you get the instant drop, but then you get a slow drop as the load continues running and battery voltage comes down, bringing inverter voltage down with it.
The trick is to have enough "room" for the load to finish running before the inverter sees 11.0 volts and starts alarming.
So in my case above, I can start with the batteries at 12.1v in the morning and see maybe 11.4v with the toaster at first, and by the time the toast pops up, I am still above 11.0v.
So IMO there is no need for over-doing the wire gauge unless you are operating closer to the margins. EG, running the MW for 10 minutes instead of just 5 minutes. You need more "room" after the initial drop with the longer run.