This is an interesting bug and without further info I can only speculate. The exterior lights get their power from the tow vehicle. In the 7-pin plug are the following wires:
+12V charge line
Brakes (not lights)
Tail lights (includes marker and license plate lights)
Left turn/brake light
Right turn/brake light
For a single brake light to come on while not connected to the tow vehicle, that line must be crossed with a 12V
source. I would say to start by looking at the umbilical plug and cable expecting to find a short between that turn signal line and the charge line. But you say this only happens when the shore power is turned on...
Do you not have house battery power available inside the trailer when the shore power is unplugged? Is the battery installed and the Store/Use switch in the Use position? The charge line is normally hot all the time if the battery is connected regardless of the shore power; with it in place and using a 12V
test light or volt meter you should find 12V
on only one pin of the disconnected umbilical plug. Do your turn signal lights work correctly when the umbilical is plugged into your tow vehicle? With a short from 12V to that turn signal line, that side brake light would be on all the time and not flash with the turn signal or 4-way flasher turned on.