It certainly looks like you may have a static drain or something turned "ON" in the trailer side. They should not drop that much overnite. NOTE I just now saw the comment you only have 12.6VDC charging your battery when shore power is on. It should read 13.6VDC.
I have one of the CLAMP-ON DC Current meters from SEARS where I can clamp around the battery cable and read how much current is being drawn when I think I don't have anything turned on. If I see more than 1AMP of DC Current being drawn I start tracking down the source of what is on inside the trailer.
Of course this could also be a bad cell in battery as well...
I suspect you have something "ON" inside the trailer drawing the power. With the DC CURRENT METER clamped around the battery terminal cable you can start pulling the ATC fuses one at a time and see which circuit drops the current flow.
Roy Ken