My 2 cents:
Get yourself a test light from an auto parts store (10 bucks). Unplug the trailer lights and test the feeds coming out of the vehicle.
The brown from the vehicle is ground so clip the spring clip on the test light to the ground from the vehicle.
Test each of the other 3 connectors individually with the pointy probe. One is for brakes, one for left blinker, one for right blinker.
If all work using the vehicle ground then you have a decent vehicle ground. If not trace back the vehicle ground and repair.
Then to eliminate or fix a bad trailer ground, buy 12 foot or so of brown lamp cord from a hardware store or big box. Run the lamp cord from the front ground connection to the rear ground connection. For the test you don't have to be fancy. Reconnect the vehicle to trailer and test the trailer lamps. If they work now then nicely route and connect the new ground wire using soldered terminals if possible. Use both wires of the lamp cord and secure with black strap ties. (never use clear because UV light weakens it)
I suggest lamp cord because it is cheap, readily available, and the multi strands is an excellent conductor. (electrons travel on the outer skin of wire so more strands is a better conductor than a single wire)
Hope that helps!