The television has a draw when "off" because it is really on "standby" but that is a different issue. You can deal with that by using more than one inverter as described or you can plug the TV into a power bar that has its own on/off switch. (when it is inconvenient to just unplug the tv, like when it is installed in an RV's cabinet)
For those small inverters that plug into a 12v socket but have no remote on/off switch, you can get a sort of adapter that has the 12v plug with an on/off switch at one end and a 12v socket at the other to plug the inverter's plug into. Now you can turn the inverter on and off by using the switch on the 12v plug instead of unplugging and plugging in the inverter.
