Looping isn't allowed. re: Battery to Power Inverter to Battery Charger to same battery.
Battery (NOT connected to the PU), to Power Invertor to PU's 110/120V main power cord would work. This is NOT a looped configuration. To keep battery charged, one might connect a solar panel (very large solar panel) charging system to the battery. From a white board diagram, this would work. Especially if all ON items within the TT (re: CO Detector, LCD TV, CD/DVD Player, etc. etc) do NOT exceed 1,000 watts load. Remember that Power Inverter should NOT be loaded over 80%. Thus, allowing some buffer room for the item's "start up" watts demand. Unknown to some, a device has 2 different watts numbers. Their Start Up number and Running number. For 12V lighting, the Convertor will convert 110V power to 12V power. Remember to include Converter watts usage in your max 1,000 watts load as well.
If doing the above, remember to turn all "large demand" 110/120V items off. re: Fridge (turn it to 12V + LP gas), Heater and large demand 110V items (like hair dryers and Coffee pots) off.