No relay needed. I think what you have done is to separate the "Brake" light function from the "Signal" function with the extra tail lights. If I have followed the post correctly. You have 2 brake lights, 2 tail lights, and 2 signal lights. All in the back. Did I get this right? I have the same setup but a different signal stat. Mine has fewer wires than yours and is not integrated. You have the "integrated" signal stat and you do not need the extra lights in the back. For a simple test, remove the wires from the second set of lights. The ones used as signal lights. Wire the brake lights to the signal wires and remove the Brake light wire. Just cover it with tape for the time being. Try the Brake pedal, got brake lights? Power from the brake switch is through the signal stat and you should have brake lights. Turn on the signal for a Right turn. one brake light should blink. Power for that light is through the signal stat and not the brake switch. Step on the brake, one blinks and the other is on steady. Power from the brake switch through the signal stat is to the steady light and power from the signal stat is to the blinking light. Note, the signal stat serves as the link to the brake lights. Not the brake light wire. It is the brake light wire that is causing the issue. When it all works, then if you want, tie the second set of lights to the brake/turn signal wires if you want.

Steve H