Okay. I figured it out. The lighted rocker switch I was using required too much voltage to power the light. I changed it out for a 12 volt automotive lighted rocker switch just to test the theory.
With the white wire attached to the #3 (Neutral) terminal, the red power wired connected to the #1 (line) terminal, and the two red transponder wires connected to the #2 (load) terminal, the LED functioned perfectly. It lit up briefly when I turned on the switch, and lit up like crazy when I tested the transponders.
Unfortunately, the lower voltage rocker switch it is too wide for the pop-out panel.
Now, I just have to find a 12 volt lighted rocker that will fit in the pop-out panel.
I'll let you know what I turn up.