I have a 2004 Ford Escape and I had the identical problem...ordered and installed the factory trailer wiring harness and the brake lights and turn signals did not work on my trailer.
The solution is to install a heavy duty flasher from Ford. The description of this item is "SF 621 : Relay - Direction Indicator" The list price was $52.22 and the local Ford dealership charged a net price of $39.24.
I also had them print out the Electrical System Relays schematic, and on this schematic, the heavy duty flasher is number 13350.
Installation of the heavy duty flasher is tricky, the light duty flasher is located under the dash on the drivers side well above the fuse panel. It is nearly impossible to get both arms in this location to disconnect and reconnect the two wiring looms that plug into the flasher module.
Hope this helps you out. email me at firstname.lastname@example.org
if you have any other questions.