First, an
inverter is not a
battery charger. It is a device which takes battery voltage (usually 12V) and converts it to AC house voltages (usually 115V AC in this country).
Some
inverters also have a
battery charger built in, mostly the larger ones. It would be plainly marked as an
inverter/charger if it had this capability.
You can tell if a
battery charger is working by measuring the DC voltage at the battery terminals. With the charger on, voltage should be above 12.6 volts and, depending on the state of charge of the batteries, could be as high as 14.4 volts.
Hard to diagnose your problem without knowing more about your setup and how everything is connected. Also, knowing about the
loads typically connected and how you use your boat would be helpful.
If you're looking for a recommendation for a
battery charger, one of the best buys on the market is the
line of Iota chargers with the IQ-4 smart charge option. These come in a number of sizes and their engineering is excellent, as is their pricing. For your 110AH battery, a 30A or 45A model would be plenty powerful and would keep your battery well-charged.
Bill