Ok got your PM and the wiring needs to be a little different.
Thinking on it !
First Pass, I am rereading it now and will EDIT anything that I think is incorrect so pls if you see something let me know.
Also, if you do not understand something, tell me, not only what you don’t understand but how I can word it so it will be less confusing.
This is wordy so I hope it is not too confusing.
I believe I would do the following:
Make this up as two circuits, one as an Off/On Switch for the OEM Trans Switch and the other for an Aux Light Circuit.
Instead of using a SPDT, I would go with a DPDT, (double pole double throw - On/On and two rows of three posts).
Make sure there is no connections between rows of Poles.
1) Locate the wire from the Trans Switch to the Lights
2) Cut the wire, extend them to the Dash Switch and attach as follows:
2a) The wire going to the lights, this should be connected to the center pole on one of the rows of poles.
2b) The trans side of the existing wire should be connected to either of the other two poles on the same row as the lights.
2c) Put some heat shrink over the extra pole.
From this point the row of poles that you have the trans switch connected to will be Row #1 and the second set or row of poles will be row #2.
What you now have is a switch with two rows of three poles with wires connected to two of the three pins in row #1.
Test #1: When you put the truck in reverse you should be able to turn the light on and off with the switch.
Now you will need to set up a standard aux light circuit and run the light wire through the sw in #2 above but connected to the pole in Row #2.
3) Setup a relay in the engine compartment.
4) Run a fused line (any interior source) to the switch and attach the wire to the pin opposite to the Trans Switch wire but in row #2.
5) Run a wire from the center pole in row #2 to pin 86 on the relay, then a wire from relay pin 85 to ground. (Get a good ground)
Another test point: Test #1: Put the truck in reverse with the key on, as you take the truck in and out of reverse, the relay should NOT click and the Reverse Light are on when in reverse. So you know the OEM reverse setup is working and that it is not actuating the relay under the hood.
Test 2: Shut the truck off, flip the switch and you should hear the relay clicking, if you don’t reverse the pole position for the power wire going to the switch #4 above; it should be on the opposite end of the sw from the trans wire row #1.
If the relay is clicking with the sw in that position you are ready to go the next step.
6) Run a fused wire from the battery to pin 30 on the relay.
7) From pin 87 on the relay, run a wire to the load/light wire and splice it in after the new dash switch; could even be under the truck but it has to be the load/light wire.
Test AGAIN !:
Test out all of the positions, flip the switch and see if it works the way you want, if the reverse lights are on all the time the wiring for the manual and auto modes are on the same end of the switch, check for it.
The center connection, row #1 need to be the light wire. and 2) the other center position row #2, needs to go to the relay.
GOD I hope this works for my sake… the switch should act as the diodes, allowing power to the lights from only one source (not that it matters) and my writing and explanations are POOR at best.
Good Luck and if you try this let me know if it works because I have done no testing, just some LOUD thinking.