Interesting that you bring this up... I was planning on tinkering on the backup lights myself!
It makes sense to me that the ESS system *could* cause errors because the Jeep system voltage will fall while the engine isn't running. The charge voltage while the engine is running is around 14.4V, but the battery voltage is quite a bit lower (probably around 12.5V with only small accessories on). The reason that voltage change is important here is because when you insert a resistor, the current that the resistor draws changes depending on the voltage.
Remember waaaay back from school, Ohm's law is E=IR.
E is electromotive force which is the voltage.
I is the current flow in Amps.
R is the resistance in Ohms.
So if the resistance is held constant and the voltage goes down, the current also goes down accordingly. Which means that when the ESS system shuts off the engine and the system voltage falls from 14.4V down to around 12.5V (that's about a 13% drop), then the current will also fall by the same percentage.
If the combination of LED current plus resistor current is near the borderline for where the lamp out indicator is, then simply dropping the current by 13% might easily be enough to trip the sensor.
There's two ways to attack this problem...
1) Figure out where the threshold in the Jeep is and make sure the total current is somewhat over that even at the minimum system voltage. I can probably help out here by using the same technique I used for the headlights. It only takes time!
2) Figure out how much current the stock incandescent lamp draws and make sure that the LED plus resistor draws a similar amount of current. This is just inserting an ammeter in series with the stock lamp and measuring the current both with the engine running and with it off. Then measure the amount of current that the LED draws. Subtract to find the difference that you need. Using Ohms Law, calculate the resistance needed to create that much current flow.
It's interesting that you're setup is getting progressively worse. It started with no errors, then started having ESS errors, now it has errors all the time. Makes me wonder if your resistors are overheating and opening up or maybe there is a corroding connection. Can you measure the resistance of your Amazon resistor? Do you know what the resistance is supposed to be?