By "charge controller" I think you meant "converter" The converter should be off the 120v supply to it when the inverter is supplying that 120 so no loop.
You can have the converter on shore power (or gen) with the inverter supplying 120v to other appliances such as the fridge. This is handy when you have 15a shore power such as visiting friend's house and don't want pop the breaker in the house. The converter keeps the RV batts up while the inverter etc are the draw on the batts.
Towing with the 120v fridge on inverter with an ATS in line, means you don't have to plug the shore power cable into the inverter like you do when parked, which you can't do with the shore cable deployed outside when driving down the road?
If you have a way now of plugging shore power into the inverter all inside the rig with no wire hanging out on the road, then you would not need the ATS. Some folks have arranged that by using an "inverter" receptacle (extension from the inverter's receptacle) for the shore power plug next to the shore cable compartment, so it is all right there.