The frig may work as a 3 way item, however, that's just the frig design. The converter does 2 things in an rv, it coverts 120V AC to 12V DC ( which runs your 12V items when not plugged into 120V) and charges your battery/batteries. Your lights, when not plugged into 120V, get their 12V's from the battery which runs through the converter for distribution. So when you trouble shoot your ceiling lights, or any other lights for that mater, don't be concerned as the propane system is not involved. The above suggestions are the way to determine what's going on. If you don't have an volt/ohm meter, you can buy one at a reasonable price from any Auto store, Good luck!
Days camping (2015)----88 days
Days camping (2016) ---110 days
2016 Wildcat MAXX 28RKX (33' TT), 2007 13' Scamp
2015 Ram Laramie Hemi
30 years RV'ing
12 different RV's