The 30A breaker is NOT something for the breakaway switch! It is one of the breakers that protects the DC circuitry in the trailer. The reference to "Ref" in the diagram is the line to the refrigerator that is hot when you are operating the refrigerator on 12V DC power when you are traveling.
The yellow line is normally the hot supply to the interior lights, not the exterior lights.
The "plastic base" you mention is the battery box under the tongue, correct? I would think you could get a replacement from J&R Repair in Ohio.
Did you buy this trailer used? It's possible a former owner did some rewiring and has mixed things up.
Where did you measure the converter output? If it was at the battery terminals, you were measuring battery voltage and your battery was discharged. You would have to disconnect the battery to see what the converter was actually delivering, and I think you might find the output was zero (0v). If so, it's likely one or two of the "hidden" inline 30A fuses inside the converter has blown. They can be easily replaced. I have a thread on this process here: https://www.hilotrailerforum.com/f28...g-repair-6454/