Re-reading your post and now understanding the AA etc. part I can see your situation is different from ours. Must have had brain fade after the long drive back from the Osoyoss rally
I agree with the previous suggestions. I use method 3 a lot, putting power from a known source directly into the part in question. Trouble shooting by isolation is a good technique. Putting power in at the plug doesn't completely isolate the problem because the problem could still be in the wiring going to the light or the light itself.
Ideally putting the power directly into the light to completely isolate the unit and show if its' working would be most helpful but the wiring connectors, at least on mine, are very difficult to access. I have an old m/c battery that I use for situations like this.
I did ask at the rally if there was a schematic for the trailer and the answer is "No".
I think there should be and, at some point, I will draw one. For example it would be useful to know where items are grounded. I have already found more than one ground and a compromised ground can create problems.
At any rate, keep on trouble shooting and hopefully you'll isolate the problem.