Re: No - running lights,only turn signals and brakes
I find running lights (side marker, not tail lights) to be the weak link in the chain--you have a mid-line splice in an inaccessible area laying on metal that gets wet and never dries. I'd rather not put the tail lights at risk.
If you are rewiring, run a dedicated line to the side markers from the front of the trailer rather than splice midline.
I removed mine. The inspector said you can't pass inspection if you have them and they don't work, but you don't have to have them (Virginia). I replaced them with reflective tape. I figure no one's going to drive between the car rear lights and trailer rear lights, being about 22' apart, plus with a big white boat on it and even the trailer guides, no one is going to miss me for lack of a 2" yellow light a foot off the ground.
Your state law may differ.