It seems evident that the amp load of the microwave is high enough that the batteries cannot maintain an adequate voltage to the inverter, causing the inverter to shut down. Not really surprising, since a 1000W (output) microwave typically draws north of 1500W to produce that much cooking power. 1500W @ 12.0v is 125A plus the inverter loss (10%-20%?), so roughly 150 amps. That's will cause a substantial voltage drop and I'm confident that the DC voltage measurement will show that. The question is why. It sounds as though the batteries can handle the load at full charge, but at anything less than 100% the voltage drops within a few seconds.
Since the 4/0 wire from battery to inverter is adequate, my guess is that the wiring between the three batteries is either too small or the connections at the battery terminals are poor, causing excessive power loss at those points.