Can't say definitely if this is your problem but this was what we encountered.
It started with the lights and furnace on at the same time. We would not notice the lights dimming down but all of the sudden they would get bright or just back to normal brightness. Then later in the year it started to do the same thing with the air on. Then also with the air off. Now the air is 110 so it should not affect the 12 volt lights. The end result was when the battery started to loose charge.
That was when after some checking with my multi meter i determined that the converter was on its last leg. Replaced the converter and all was good. Also took this opportunity to see where all the wires went - what fuse to what light or appliance and what breaker to what outlet - then labeled the new converter accordingly.
As i stated not sure if this is your answer but it was mine.