Yea it would but your going to need to create an auto ac tranfer switch and have power split from an inverter and regular wall power. Also if you don't have an understanding of things like resistance, amperage, voltage and and good wiring skills I don't recommend doing a project like this as it can be a huge fire hazard and potentially life threatening if you do something wrong. My backround is a proffessional auto mechanic that is going to to school for electrical engineering. And even I made a mistake by adding a dc power supply that was too large for the dc relay and it had to me replaced, do to it heating the relay coil to much. I had to replace it with a dc power supply that was eqivalent to aprox 150mA. As this was more appropriate for an average relay amperage draw.
Now that said, the project is relatively simple electrically and can be quite fun, but does require safe disipline as all electrical projects do. And I encourage my warning not to scare you just to realize what you may be getting into. I just ordered my 300watt inverter for my heater and and internet router. I'll be adding them also. I'll post more photos when I get parts and finish further.