That is strange - The TRAILER SAFETY LIGHTs are powered up by the TRUCK through the 7-way connector. Has nothing to do with the trailer BATTERY setup.
I use one of these 7-way testers on my Ford 7-way setup. If this works normal when plugged into the 7-way Connector then I am assured the truck side is OK...

The only connection from the 7-way connector to the trailer wiring would be the 12VDC lead coming from the truck start battery used for trickle charging the trailer battery when in route.
ETRAILER.com is a great place to look at some typical wiring diagrams of the trailer safety wiring etc...
All of my 2010 F150 truck safety signals are separately fused at the truck fuse panel and most everything going back to the 7-way connector is controlled by the truck ignition key being ON...
Have never seen this problem before...
Roy Ken