My situation is different, but might be useful (or interesting). I own a 1963 C10 that has nonfunctioning reverse lights. I’m not sure if they were put on at the factory, or were a dealer add on. The reverse light lamp buckets are on the truck below the tail lights. I have a three speed on the column. The steering column sleeve has a reverse light switch that I show in this picture. So my truck at one time had functioning reverse lights. I suspect it was a dealer add on. I have no idea where a reverse light switch would be located on a 4 on the floor truck. None of my info directly answers your situation, but I thought I’d chime in.
