Relocating the battery charger help

I thought the charger/inverter "kicked in" to help run things (when on shore/generator power) if I started to draw more than the battery could put out. Hence the reason to run cables from the charger to the inverter. Or do I have this completely messed up in my head?

Also - I didn't even think about from the batteries to the DC power distribution box. The factory battery cables will be ok for that? Would it be worth the extra $ to swap out with something "bigger"? If yes, what size?