The real simple test is to use a multimeter and read the battery terminals with the shore power turned off. This is assuming that both batteries are 12VDC batteries connected in parallel. If the battery is fully charged it will read 12.6-7VDC. Then turn ON the Shore Power and reading the same battery terminals should now jump to 13.6VDC. This is telling you the connection between the converter/charger and the battery terminals are good. It is also telling you the converter/charger unit is working.
If you do see any increase in DC VOLTAGE when you turn on Shore Power then obviously you are not making connections between the converter/charger and the battery terminals. Both the battery connections and the DC output of the converter/charger unit should feed the 12VDC power distribution panel where all of the 12VDC fuses are located.
There is a couple of things to check. You should find a in-line fuse real close to the battery terminals. Check this fuse. You also have two fuses on the 12VDC distribution Panel labeled REVERSE POLARITY. These two fuse will be be high current values. Make sure they are not blown. These fuses and mechanical battery cable connections are about the only things between the output of the converter/charger and the battery terminals. You have already said the ceiling lights etc work just fine when on Shore Power so this tells you the converter/charge unit is working OK.
If your trailer has a battery disconnect switch installed it must be connected to charge your batteries.
Shouldn't be too hard to tell where you are not getting the 13.6VDC coming from your powered up converter/charger unit and the battery terminals.
just some of my thoughts
Roy Ken