Not to say how your boat is wired but mine has a seperate 110 loop running through it. and 6 plugs around the boat (for the TV -coffee pot & such) the 110 loop is NOT tyed into the rest of the boat or the batteries.. off the 110 loop the battery charger
has a breaker tyed into it..
Now my inverter is tyed to the house bank and only works when I turn it on and run a jumper plug
to the 110 loop.
A good way to check is to un-plug the 110 shore-power and see if your TV works..
If it does, you've got an inverter in the system. Some systems, Like Xantrax make an inverter charger
combined.. works like a charger and when shore power
is turned off, it works as an inverter.. nice system but very pricy.