A 900 Watt microwave is almost certainly 900 Watts of output cooking power, but it draws 1400 Watts of power to produce that 900 Watts. Inside the door or on the rear of the microwave there will be a plate that tells the power consumption of the microwave, which is different than the output cooking power.
In either case, the inverter isn't the problem. The problem is the batteries. The batteries are either too small to handle a really large amp draw like 1400 Watts (1400 Watts divided by 12 volts = 116.66 plus 10% for the inverter loss is 128 amps), or their capacity is diminished to the point where the voltage simply drops off too much and too rapidly when such a high amp draw is applied. If the batteries are hybrid batteries (truck or marine) there ya go, as those plates are relatively thin, and are not designed for sustained high amp draws.