Your lights should be on the 12VDC (battery-connected) system and the A/C (compressor, at least) is on the 120VAC system. There should be minimal interaction between them - only connected at the battery.
However, is your battery the same vintage as the RV? If so, it may be nearly dead and your lights are really being only run by the 12VDC being produced from the converter for battery charging. Thus, when the AC sags from the air conditioner startup surge, maybe the inverter charge voltage sags too and there's no battery to keep it up.
This is why most converters will fault if there is no battery.
If you're saying that the lights become brighter than normal after the A/C gets going, that would be strange because that kind of thing is usually a symptom of bad neutral bonding in a split phase (household) system and you don't have that in an RV.
2015 Rockwood Signature UltraLite 8282WS Platinum, GY Marathon LRD, TST 507RV TPMS
2005 GMC 2500HD CCSB D/A, Curt E16, Prodigy P2, Garmin RV760LMT w/BC-20 b/u cam