I would lean more toward a fault in the tether between RV and car or at one of the connections before the diodes.
I don't think of the diodes as being involved with the backup lights on the RV.
Maybe do some sorting by disconnecting the car, verify the RV wiring is working right. It could be as odd as something like a rock mashing the connector right on the RV but I suspect not. Just good to check and find which way to move.
If RV all good without tether, then connect only the tether before adding the car and retest.
If RV and tether are both working without car connected, then you may need to move to some method of connecting only the ground from tether to car and then a small jumper on each section of the tether wires like one turn signal and then the other as you move through the connections one at a time until you find which work and which fail.
This is a place where some jumper wires with alligator clips on each end can help a lot as it helps avoid running out of hands to hold things! Small clips that can fit into the connections are nice??
Do you know if you have a drawing in the lid of the RV trailer connection? That would be a big help on ID of which wires in the tether serve which function.
If not labeled, this is a drawing that is considered "standard" but not always used! Check before trusting!