From all the various wiring diagrams I've seen for these trucks, I think it is safe to say that all 3rd generation Rams have a FCM that directly drives the headlights, brake lights (except Center High Mounted Stop Lamp) and turn signal lights, which are the only lights that will trigger a "Lamp Out" warning. I was unable to determine exactly how the FCMs in our trucks actually implement the lamp out determination, but one very common way is by balancing current flowing through two bulb filaments.
The controller likely compares the currents between two different bulbs, for example, the current drawn by the left front turn signal bulb with that drawn by the left rear bulb. If the current through both bulbs is the same (regardless of specific level of current) the system would do nothing. If, however, the current drawn by one is much greater than the current drawn by the other, it might signal a bulb being out. This test is much simpler to implement than determining an absolute current level and comparing it to a reference because the actual level of current does not matter regardless of voltage level. Since the voltage in our trucks varies dynamically, the specific level of current flowing through a bulb would vary from time to time, but at any given voltage the currents being compared (left vs. right; front vs. rear) would still match. Additionally, if comparing actual current, each different bulb system would have to have its own reference value (e.g., expected high beam current, expected stop lamp current, etc.), whereas by comparing the two currents, no reference values at all are required.
I could be wrong, but I believe that the 3rd generation Dodge Rams use current balancing. The following statement from the '04 FSM reinforces my belief:
Each time the cluster receives a lamp out indicator lamp-on message from the FCM indicating that an inoperative headlamp (low or high beam), turn signal lamp, or brake lamp (excluding Center High Mounted Stop Lamp [CHMSL]) circuit has been detected, the lamp out indicator is illuminated.
Note that each of these monitored circuits are "paired," either left to right (brakes, excluding the CHMSL; headlamps) or front to back (front and rear turn signal lamps), and in their factory condition each of these lights consist of a single filament driven directly by the FCM. (Note that while a 3157NA is specified for the front park/turn signal and a 3057 for the rear tail/turn signal, the turn signal filaments of these two bulbs are 27 watts and 26.9 watts, respectively, so they still nevertheless "balance.")
Whew! All that for my next suggestion.
Have you ever considered using LED 3057 bulbs for your rear tail/turn signal lamps? I don't guarantee it would work but it seems to me it might be worth a try.