Stuart, you are being led down many paths here. All, I am sure, are with the best intentions. Let's approach this methodically and logically.
Please correct me if I have misunderstood any of the facts you have given here. And from the other threads you have started about this same issue.
1. You state that when you are connected to the battery the lights work. If that is so then the battery in-line fuse, the converter main battery fuse, the reverse polarity protection fuse and the fuse for the lighting circuit are all good.
2. When you disconnect the battery and connect to the shore power your lights do not work. Your 120V outlets do work. Then the main AC breaker in the converter is OK. You stated that you have reset the breaker that is for the converter/charger. The breaker COULD be bad but that is highly unlikely. Now, there is probably a fuse located on the circuit board of the converter electronics. This is not typically a user replaceable fuse.
If you feel comfortable doing so you might remove the circuit board from the converter and see if you can locate this fuse. If it is, in fact, blown there was likely a failure of the circuit board itself that caused it.
Now you have two choices: A) Replace the converter. B) Have the circuit board repaired.
I hope this has cleared up the possibilities that exist and what is not possible.
Chuck - Sparta, TN
2012 Jay Flight 22FB, 2 x Honda EU2000i
2013 GMC Yukon XL Denali AWD
EDUCATION is what you get when you read the fine print.....
EXPERIENCE is what you get when you don't.