When you say all the other exterior lights are working, I'm assuming you mean the yellow and red clearance lights at the top of the TT??
If so, then your problem is NOT the tow vehicle (at least not for the tail light circuit, which also powers the clearance lights). I suspect you've lost the common ground associated with the tail lights and brake lights. On my Roo 19, you accessed these by removing the thin plywood panels that FR installed on each end (left and right side) inside your rear storage compartment.
As I recall, the wiring for the rear top clearance lights drops down and is tied into the tail light wiring. There will be two separate power wires coming from the front of the trailer, one for the tail lights and one for the brake lights. But there is only one ground wire. But this ground wire has to tie into the ground wires for all the clearance lights, tail lights and brake lights in the back of the TT. This ultimately requires a number of splices into the ground wire. I suspect that the ground wires for the tail lights and brake lights may have been tied together, and then to ground, but that connection has been broken.
Hope this helps. Good luck.
1988 Coleman Sequoia - popup (1987-2009) - outlasted 3 Dodge Grand Caravans!
2012 Roo19 - hybid (2012-2015)
2016 Mini Lite 2503S - tt (2015 - ???)
2011 Traverse LT, 3.6L, FWD
2009 Silverado 1500 Ext Cab, 5.3L, 4x4, 3.73
2016 Silverado 2500HD Dbl Cab, 6.0L 4x4, 4.10