That was a good explanation for sure but I have found that many older trailers and smaller ones have no real converter section at all.
It may be that you are simply charging the battery with the charge line of the vehicle through the seven pin connection to the trailer when connected?
Is there any reason to think you are doing anything different plugged into AC other than powering the AC outlets directly?
You mention that you think it should power the furnace
better but you do not say if you have done so and if it does?
My first trailer had no electronics in it at all.
If you can not find a converter/charger maybe there is none.