Hi ....
Sounds like the wire that supplies the 12VDC to the trailer from the
tow vehicle is shorting to the wire that sends juice to the running lights
of the trailer. This would allow the trailer
battery light the running lights.
I would start with the plug that plugs into the trailer by the hitch when
troubleshooting this.
BTW, disconnect the trailer
battery until you get the problem solved.
Larry H