YOu need those tests but let me run down some possibles
First, there are basically two ways to hook up a converter.. The house side, or the batery side, of the battery disconnect switch or solenoid.
If it is hooked to the battery side, then it will charge the batteries EVEN WHEN THE SWITCH IS IN THE DISCONNECT/STORE position
NOTE: If your converter, is also an inverter.. then it is wired this way.. Like my Prosine 2.0.
Second, for the purpose of this post there are two types of converters.. One is like my Progressive Dynamics Intella Power, it's wired in parallel with everything else, has just two 12 volt terminals,, One Red, One black (Positive and negative) and that's all (Well it also has a120 volt cord)
The other, is like the Magnatek 6300 line.. This one has two 12 volt POSITIVE outs.. Generally the other one is blue.
IT has a relay inside it.. If running on shore power this normally closed relay will pull open and let the power supply portion of the converter run the lights and other non-senistive loads.. While the battery charger portion charges the batteries and also runs things like the radio which are "Power line noise senistive".
IF that relay fails, (open) then you won't have lights (But the "House" radio may still work) when you do not have shore power.
There are also fuse and circuit breakers that can affect you again depending on how it is wired.
THe solution is to chase the power.. I recommend a test lamp rather than a meter (Easier to read quickly)
I took a license plate lamp socket from my local auto parts tore, This socket takes an "Edge type" or "Blade type" Bulb (has a flat blade that goes into the socket) though any kind will work so long as the socket is for a single element blub and has TWO wires coming off it.
I put a test clip on one lead,, extended the other lead about 3 feet to a 2nd test clip.. This is my test lamp.
Bright or Dark is all I need to know.
Home is where I park it!