I understand your confusion....
Inverters come in 2 by 2 by several types We will discuss only the two tows
True Sine/MSW.. You already made the proper choice, so short discussion here.
In-Line (Inverter/converter) Stand-alone (Does not charge)
I think your 600 Watt is of this type.. A Stand alone... If so let me describe a very expensive device I saw in an electronics store some time back.. A Power Line Conditioner.
Source power (Mains)---Converter---battery---Inverter---Protected loads
So use the inverter to power the TV.. Feed it off the battery, Let the converter feed the battery,, A very good set up
Works great.. I do that when I'm on flakey power sites with my setup.