Shore power and generator power are both 120 volts. That power needs to go into your converter to produce the 12 volts needed.
Put your meter on the converter output terminals. You should show 13.6 if the converter is working.
Now, if that is the case, you need to get that to the battery. It might be the positive feed to the battery, or the negative.
The best way to check is take your jumpers and connect to your battery positive and battery negative. Take the other end of your jumpers into your rig, and with a strong wire, first connect the negative to the negative output. If your lights all of a sudden get bright, you have a loose connection in the negative wire.
If nothing happens, remove the negative jumper, and connect the positive to the converter positive. If the lights get bright, you have a loose connection in the positive wire.
If that doesn't work, connect both positive and negative. Let me know what you find.
In my case, the negative heavy feed wire connection to the battery was loose in the terminal strip at the converter.
Also a good idea to tighten all the wires in there. Check the battery terminals for the negative and positive feed wires.
The alternator charges because that is a 12 volt connection to 12 volt connection.