well you would just keep the whole thing 12v and just supply a power brick adapter like this: http://www.amazon.com/dp/B00IK8ETNI?psc=1 so it will convert home wall power to 12v DC while at home. So basically if you intend to make it mobile keep it 12v dc and no inverter means that there is less draw on the battery as well making it "safer" for the car 's electrical system which is probably only rated to maybe 10A max.
Edit: At least thats my $.02 and what I've found when I pursued building a dual purpose home/mobile rig