Why won't 2000w inverter run 900w microwave?

A double pole double throw switch will let you toggle between the inverter and shore power, the battery is not involved (aside from powering the inverter) and it is a two way switch. Which is not your problem - your problem is probably wire size. A 2000 watt inverter can draw 170 odd amps at 12 volts, which requires a 1/0 gauge size wire in relatively short runs. Even the microwave, at 900 watts device power will draw around 90 amps and need around a 1 gauge wire. You probably have a lot less than that, so look at that first. The check, as noted, is to read the voltage at the inverter terminals as the microwave is trying to run. They surge at first, and probably that surge is causing the inverter to go into low voltage protection.

Brian