An advantage with MSW vs PSW to run the MW (assuming it will, although slower) is it draws fewer amps.
Eg, a "1000w" MW might pull 150 amps with PSW but 120 amps with MSW. I ran some tests a while back that showed to "cook" the same thing at the longer time, the MSW used fewer AH. The A being a bigger factor than the H.
Another result is at the lower amps, your voltage drop is less, so you don't hit the inverter alarm on low battery when you would with PSW. You could use fatter wire with the PSW though.
Previous threads on this mention the Panasonic inverter MW which has the advantage of running at lower power for lower settings instead of always at full power but more time off than on for lower settings.
That gets you the lower amps and AH too. It did come out that those MW don't like MSW, so you can't have both to get even lower amps but more time. That was a few years back, so it might be that those inverter MWs can run on MSW now, don't know.
