Thanks for the very detailed troubleshooting steps. I'll go through all these when the problem forces me to, or I have the time and energy to do it before that (I can always drain the battery on purpose if needed to simulate the failure condition).
Couple of things to know about this situation, though:
Due to the climate out here in CA where I am, there is very little corrosion anywhere on this van. I actually removed, inspected, and re-soldered a couple of joints in the main relay a few months ago, and my 17 year old 230k van's main relay circuit board looked 10x better than any of the photos of the repair. It was a pre-emptive check/repair just because a few others were finding problems with the main relay.
I think I've pretty well narrowed this problem to be related to battery voltage. I know sometimes flaky electrical connections can be voltage-dependent, but here I'm observing reliably that 12.0V is too low and 12.3+V is OK, which is a pretty close range, so I'm not really expecting that wiggling wires will provide the difference required to satisfy the immobilizer with only 12.0V at the battery. But I'll do these tests anyway.
I've got no problem modifying the car to bypass this if it is a good, reliable solution. For example, doing the main relay hot-wire, but with a switch in there so I only do it when needed. I'd just label the switch "initiate self-destruct sequence" so none of the car thieves trying to take my sweet old ride would use it to circumvent the immobilizer.