If that one light is on "bright" all the time (the slight pulse you describe is caused by the bulb getting extra current and then not) you have a short circuit some where. The short is allowing full voltage to both elements of the bulb continuously, so, somewhere you have a 12v wire that is hot all the time contacting the wire for the second element, that which controls the brake and turn signal brightening. The devil will be in the details...read it is time to break out the VOM and start doing some testing. You'll have to start by getting the schematics for a 7-pin trailer harness so that you'll know which wires control what and then you'll have to start at the plug on the tow vehicle and start working back from there. If both plugs are good, then you'll need to trace the wires to the battery disconnect switch area; that's where the junction box will be located where the individual wires branch off of the main plug cable to go to each light.
Tracking down these types of wiring issues will test your patience, but, they can be found if you have the patience to keep analyzing the wiring. I once had a water heater that wouldn't light up and would have sworn that it was the controller board. Bought a new board and it still wouldn't light. Turns out it was splice that had gotten loose inside a wire nut. Tightened the wire nut and all was good. Takes patience and perseverance.
My 2 cents, your mileage may vary...
2 Cats; J-Lo and Ragamuffin :R
2014 Thor Tuscany 40RX DP
2011 Ram 2500 Longhorn CTD HO
2011 Keystone Cougar 318SAB (now gone)
2008 FunFinder X 210WBS (Sadly gone)