It seems everything is doing what is expected. When you plug your trailer into a 120VAC Shore Power Connection then the on-board Converter/Charger will charge your battery. The 12VDC output of the converter/charger is most likely a single mode charger and will produce a constant 13.6VDC DC Charging voltage to your battery bank. This will eventually increase to 13.6VDC while being charged.
This appears to be normal for your converter/charger.
This 13.6VDC however is right at the point of boiling out your battery battery fluids and the batteries fluid levels should be checked at least every two weeks if continuously left plugged in to shore power.
Replacing your converter/charger with a smart mode charging technology unit will resolve this situation. If you are worried about not being able to look at your battery fluid levels on a scheduled basis then I would disconnect the negative battery terminal when being stored.
just my two cents
Roy Ken