There would be a few things to check here. The system is more complex than an older system and may require the use of a factory scan tool to diagnose this concern. You said you were getting 3.94 volts on the gry/red wire. This circuit should have battery voltage any time the key is on. The same circuit and fuse also powers the left rear turn, license plate and right rear park stop lamps. The same relay that powers that fuse, also feeds power to the courtesy lamps and other ciruits through various fuses. If these are also a concern, there is high resistence in that circuit or relay, but it doesn't appear to be the case and I wouldn't suspect just a low voltage concern to just the back up lamps and not the other circuits. Are you sure you have your neg test lead on a good ground? Since you said the right side lamp illuminates, you probably do not have a voltage concern if the lamp lit brightly.
Now to the grounding side of the circuit. The backup lamps are turned on (grounded) by the rear electronic module. It receives it's signal from the pcm over a high speed data line telling it to turn on the backup lamps. The pcm gets it's signal from the trans range sensor. It is possible for the trans range sensor to be faulty and you may be able to test this yourself.
Using your volt meter, attatch the pos side to the battery pos terminal. Attatch the neg lead of your volt meter to the lt blu with black tracer wire at the tr sensor. Set your emergency brake. Turn the key on, leave the engine off. Place the shifter in the reverse position. Your volt meter should read battery voltage. If it does, My suggestion is to have it diagnosed by a professional with a factory Ford scan tool. If you show no voltage or little voltage, place the negative lead on the gry/red wire at the tr sensor. your meter should show battery voltage. If this time you have voltage, and the first time you didn't, the tr sensor is either faulty or improperly adjusted. If you have no voltage, I'm back to recommending a professional checking your vehicle.
Also verify connector condition at the tr sensor and that all pins are clean and fully seated into the connector. You may also want to give each wire a slight tug to verify they aren't broken inside the insulation.
Good luck and have a nice day.
You will need to backprobe the wires at the connector. You may be able to use a small needle or pin pushed into the backside of the connector for the circuits I explained above and touch the meter leads to the pin. You need to make sure the pin is making good contact to the wire within the connector or the test will be invalid.
You will need to remove the connector to install the pin. I suggest using a couple pins at the same time, not allowing them to touch each other or anything around it, so you don't have to keep unplugging the sensor.
Good luck. I wouldn't mind knowing what the fix was when it gets done, if you don't mind.