Even though it is embarrassing to admit how long it took me to catch on to the problem here, I thought I would document it, in hopes that it might help someone else.
My son drove this Marauder one day, then drove it home for lunch. On his way to the house, the battery warning light had come on. So he parked it and took another car back to work.
When I got home, the battery voltage was reading in the high 11s. But it would crank, so I started it up. Even revving the engine up did not kick the alternator on, at least as evidenced by the voltage reading, which stayed in the 11s.
I happened to have a new alternator in my stash of Marauder parts, so I swapped that one in. Now when I revved the engine, the voltage across the battery was about 14 volts. Great, all fixed, I thought.
So, I didn't want the alternator to have to charge the battery back up, so I removed the negative terminal and charged the battery with my plug-in charger. Just after taking the charger off, I was at about 13 volts across the battery. So I hooked up the negative terminal and went to start the engine. Nothing. No headlights. Not even any "dome" light. Just nothing.
I scratched my head, undid the negative terminal, then put it back on, tightening it a bit tighter. Now the lights all worked and the car started right up. But after it had run for about 30 seconds, the battery warning light came back on and stayed on. I was puzzled, and didn't feel very confident that the new alternator had addressed the root cause.
On the weekend, I took the car to AutoZone where they tested the old alternator, as well as the battery, starter, and alternator in the car. Everything, even the old alternator, checked out as good.
Then on Monday, the car was driven about a mile and parked back at home. The next time we tried to start it, we got nothing again. No cranking, no lighting. But the voltage across the battery was about 12.4 or 12.5 volts.
Once again, I removed the negative terminal from the battery, and re-attached it, tightening it securely. Once again, everything worked. But in 30 to 45 seconds, the battery warning light came on again.
(For those of you keen to guess, stop reading here and make your diagnosis. It's pretty obvious, actually. Answer is below.)
So here was the cause: Corroded battery terminals and cables. Especially the inner surfaces of the cables which contact the terminals. I cleaned all the corrosion off, and now everything works properly. No more battery warning light. It runs better, too. Before, the engine seemed to be "breaking up" when getting on it hard. Now it is once again very smooth. Apparently, so little current was able to get through the corrosion at the terminal that the demand of the ignition system overpowered the reduced capacity of the electrical system.
The only thing that pisses me off is that I should have cleaned the terminals and cables when I first got the "no lights" condition just after charging it.