All of your interior lights operate on your COACH battery which is generally located on the right (passenger) side of your RV. The power from the battery goes to the 12-VDC fuse panel, and from there it is distributed to different 12-Volt systems. Often, if the COACH battery is not fully charged then there will not be enough power to the longer wired runs.
I would check the COACH battery voltage, using a multimeter. If the reading is 13. 5-VDC or 14.5-VDC then the battery is charged or it is being charged.
But, if it is only around 12.5-VDC then the battery needs charging, which is done by your CONVERTER. The Converter is powered by your 110-VAC which comes from your external power source, generally from your campsite power box.
Check these things to find the most likely cause of your problem.