@iamretired it does go through the transmission range sensor. The Transmission Range (TR) sensor , which is located at the manual lever on the transmission is an information sensor for the powertrain control module (PCM). Among its functions are those normally handled by a conventional Park/Neutral switch: it prevents the engine from starting in any gear other than Park or Neutral. and closes the circuit for the back-up lights when the shift lever is moved to Reverse. The TR sensor also contains the sensing circuit for Neutral in 4X4 Low range on 4WD models.It should be located on the shift lever side on the transmission.
Looks like if you place a jumper between the LtBlu/Pnk wire and the Blk/Pnk wire it'll bypass the reverse sensor and should turn your backup lights on. If so, you know its your TP Sensor that has failed.