Hi,
A 2000 watt pure sine wave inverter would be needed. My microwave draws 1580 watts and sometimes did not start on a 2500 MSW inverter. I occasionally would have to have another load on first. Since I moved to 3000 watt PSW there are zero issues with starting.
A 1500 watt may not start the 1380 watt microwave reliably unless it is one of the Panasonic units that have low power settings that actually work. Most microwaves cut back on the heating power by running the unit flat out--but for differing slices of time.
i.e. power setting 1 might be 1/10 of Magnatron on time per second during run time.