Close to my setup - YES it can easily be done - Ours Works Great
1200 watt inverter microwave used many times during the day but no coffee pot Just yet
Our 2Kw sine wave inverter easily powers the microwave for as long as needed
The other half of the equation the batteries, charging & Metering / monitoring
200amp/hr LiFe battery bank that is dedicated to the inverter and heater blower motor no other house loads. A 220amp/hr lead acid battery bank for house loads. 600watts of solar and when cloudy for extended periods a 4kw generator with both a 100amp and 75amp converters
Build yourself a nice system and you will enjoy it for years to come