When you're connected to shore power, your converter is powering the 12 V appliances in your trailer. So the fact that the tank monitor says 13.5 V when connected to shore power seems fine. I'm guessing that you haven't pressed the use/store switch to use. When it's in the store position the batteries are not connected and therefore not charging via the converter. To charge your batteries, connect to shore power and make sure the use/store switch is in the use position.
Since you have a BlueSky IPN Pro, I'm guessing you have solar. If setup correctly this monitor always gives you the battery voltage directly - regardless of the use/store switch. So if it's saying 8.3 V, I'm guessing your batteries are probably shot. Get yourself a voltmeter and measure the voltage directly across your AGMs to confirm that they are at 8.3 V.
2015 30' Classic, "Chez Nu"
2014 RAM 2500 w/Cummins Diesel
ProPride Hitch, 400 W Solar