Trailer running/stop/turn signal lights are completely separate from trailer battery and it's circuits. I think you have a problem with your trucks wiring or with the connector between truck and trailer. If none of the lights are working, it could be you don't have a good ground.
1988 Coleman Sequoia - popup (1987-2007)
2012 Roo19 - hybid (2012-2015)
2016 Mini Lite 2503S - tt (2015 - ???)
2011 Traverse LT, 3.6L, FWD
2009 Silverado Ext Cab, 5.3L, 4x4, 3.73
2016 Silverado 2500HD Dbl Cab, 6.0L 4x4, 4.10