I'm a little dense and it has taken me a little while to decipher your post. I get the impression that what you are calling rear running lights are what we call tail lights. The thing that has me confused is the use of headlights and the clear lens on the light you show in your post.
It may be a little confusing but if those are red leds in that light, then I would bet that only part of the LEDs are lit, or, those resisters are used to allow them to work at a lower current flow and when you apply the brakes, either all the LEDs come on or full voltage is applied allowing them to signal you are applying the brakes.
And yes that can be done. Those resistors and diodes are there for a reason. I would remove and disconnect the lights in question and check for electrical operation of the wiring. If power is supplied when appropriate and the grounds are good, get new running lights. It doesn't seem to me that it is a ground issue and why did the trailer stop blowing fuses?