My trailer was prewired for a back-up camera. I'm pretty sure the prewiring just involved splicing the wires for powering the camera into the supply lines for the running lights (no special circuitry added). I installed the back-up camera using the preinstalled wiring (cutting off adapter that was specific to the Furrion camera and splicing in the wires attached to the camera) and the camera has worked without issue and with no noticeable dimming of the trailer lights.
I'll check to see what the current draw of my camera is but I did not notice any heating of the wires as you've described. The current draw for your camera should be either shown on the camera or in the documentation that came with it. As a point of reference the current draw on a popular model Furrion camera is listed as less than/equal to 300mA so if the current draw for the camera you have is similar I would think it should not cause a problem with the running lights.
