If the battery has a shorted cell you will not get full voltage even when the converter is running.
The AC has nothing to do with the 12volt system since it runs directly off the 120volt shore connection. (same for microwave)
Lights (most of them), jacks, slideouts, and control boards for fridge and heat run on 12 volts. Shore power through converter to battery to device. Bad battery or bad converter interrupt the flow.
With the battery disconnected, the converter should be putting out 13 to 13.6 volts measured at the battery terminals. The disconnected battery should show AT LEAST 12.4 volts across the terminals under no load.
From what you have said, I'm starting to suspect the converter.