Just be aware that with the converter turned off your battery is not being charged and will probably drop below 12.0VDC pretty soon. Continuing operating the battery below 12.0VDC (approximately 50% charge state) usually does harm to the battery requiring it to be replaced as well...
You can hook-up a portable battery charger to your battery setup if you have one.
You need to make DC VOLTAGE measurements to determine what is going bad...
You should read 12.6VDC across the battery terminal when not on shore power and the battery is fully charged.
Turning on shore power which in turn turns on the on-board converter/charger should read 13.6VDC on the battery terminals. (Your ceiling lights should get brighter when you do this). If you have a smart mode converter/charger installed on your trailer then the DC VOLTS OUTPUT will normally be at 13.6VDC or sometimes 13.2VDC. It will produce 14.4VDC when in boost charge mode for around three hours but for the most part will read 13.6VDC when it is turned ON. Some model converters/chargers will go to above 15.0VDC for about '15 minutes' every 20 hours doing battery maintenance. Mine does the 14.4VDC when doing this maintenance mode. A smart mode charging system should not boil out battery fluids but you should always get in the habit of checking battery fluids. I check mine every two weeks or so.
Roy Ken