[Fulltiming]

Alex,
Your volt meters show voltage on both L1 and L2 so, it appears, power is coming in on both AC legs. Have you checked all of the electric outlets to see if they are all live? Although the lights not being illuminated on L1 could indicate a problem, they can and do burn out. They are neon lights and I have replaced over half of them on my coach. On my 1992, the inverter was wired in differently than any other RV I have owned. The inverter didn't have a built in transfer switch and did not incorporate a battery charger. The inverter was turned off and on through external relays and I had a separate battery charger (single stage battery boiler) which was located in the bay with the transfer switch and the power cord. I had the inverter replaced with a larger pure sine wave inverter with a built in transfer switch and built in multi-stage battery charger. A significant amount of rewiring was required to bypass the original external relays. If your inverter is on L1, those relay circuits could be an issue in your case.

Richard is correct, there are dangers in getting shocked if you are not comfortable with electrical diagnostics.