First, welcome to the forum.
Do you have a inverter like you indicate in the title or do you have a converter like you indicate in the post? I'm thinking you may find you have a converter.
Follow the wires from the battery. They should lead you to the converter. That converter may have fuses on the end of the case. They're there to protect the converter if someone accidently hooks up the battery cables to the wrong battery posts.
While hooked up to shore power and using a volt meter, see if you have voltage at the ends of the battery cables when disconnected from the battery.
At the same time, check the voltage of that battery without the cables connected.
Also, you may have a dedicated 110 Volt AC circuit breaker for the converter. If you do, see if it has tripped. If it tripped, you'll need to shut it off then back on to reset it.
If you do find the converter, tell us the manufacturer and the model.