Hmmm...11.4V (if correct) for a battery that has been charging suggests a shorted cell in the battery. It may be that this is the main part of your problem, but there's still a lot of voltage drop (indicative of a poor connection or quite undersized wire somewhere) between the converter and the battery.
Do check with your voltmeter that the battery monitor is giving you a correct voltage reading. If it's reading 10 or 15 percent too low you may be chasing ghosts.
The generator switch would not be your problem here. It's not related to the 12V system, or at any rate to the connection between the battery and the converter.