Makes sense, I was just wondering if the boat had been refitted to run dual voltage (12V and 24V) systems. Being that your installation seems to be OEM, I would concur with Don Lucas recommendation that you procure the boat owner's manual and get familiar with the use of the system as designed. Once you realize its virtues and shortcomings then you can start making changes to suit your specific needs.
This should be straight forward
Firstly, as suggested go to the hunter site where the manuals are located
Charging the batteries from the engine has nothing to do with your inverter. The inverter converts 12v to 120vac (or 220)
On your boat you will likely have a VSR relay which shunts the house to the engine start.
I assume the house batteries are charging fine from the battery charger, or you would have mentioned that.
Is the engine battery charging when the shore charger is running? I suspect not. If I am correct, then the VSR is likely defective