All I was referring to was the path the electricity has to draw a ground connection through. i. e. the only way is through the bulb filaments if there is no effective ground. The path through the filaments is a high resistance ground.
If everything else is correct, I still think you will find a faulty ground connection. The connections maybe solid, but is the surface under the connectors clean or the other ground points clean? Run a jumber from the vehicle ground to the bulb housing, and I bet you'll find that particular lamp burns just fine, possibly all of them if they share the same ground point. Do the lights bolt to the frame of the trailer and do they provide the ground? If so, the metal surface under the housing needs to be shiney bright, or the paint/rust if it exists, makes an excellent insulator.
Are you fighting two problems? i. e. the turn signal flashes both, so are you on the tail light circuit by any chance? I believe the other when you hit the brake is related to the ground issue. The tail filaments will be noticeably dimmer than the brake filaments that are used with the turn signal. I'm sure you knew this already, but had to throw it out anyway.
Good luck; chasing electrical gremlins in vehicles can be maddening, but a systematic trouble shooting plan will weed it out.