Do you know that it is an LED for illumination? It looks to me like you maybe have an incandescent bulb (or something that doesn't care about polarity) and you have the POWER and GROUND connections reversed.
It's possible.
I'm not certain if the bulb is LED or not. I assumed it is LED because it works fine on 12v DC and usually other type bulbs in these illuminated rocker switches require much higher voltage like (AC).
If it is connected incorrectly, how does one explain that it works perfectly fine until I connect the external battery?
When no external battery is connected......1). The rocker switch correctly turns the circuit On and Off and I can read 14.4v at the alligator clips (NOT attached to a battery)
2). The LED comes on as expected
3). Placing the rocker in the OFF position, the LED turns off and the circuit powers down.
It is ONLY when I connect an external battery to be charged that problems develop.
It may also be significant that if I do not connect the black wire that powers the LED, I can then connect the alligator clips to the external battery to be charged and it turns the circuit on and off normally and as expected (just without any LED)