The one last thing that I find odd is that the converter is not powering the fridge directly but using the battery as a buffer (From what I can make out but maybe I missed something). The reason I find that odd is because on our coach and every TT we have owned, whether there was a battery or not, the fridge was powered directly by the converter when on shore power.
The only 12v items I have ever encountered that required the battery to be installed were high draw like an electric tongue jack or hydraulic slide motor and entry steps.