This is such an interesting conundrum! That narrows the problem:
1) it stops when you are *decellerating*
2) at *20 mph or less*.
3) *everything* goes out, lights, engine, all
4) starts back up in a few minutes; *full power*.
Here is my less-than-wizard take on this. I bet that there is some heat-related issue somewhere in this, if everything works in a few minutes. Something cools and contracts somewhere, and it might be electrical resistance heat, not engine heat. The slowing/stopping effort makes something move forward slightly/or the bearing down on the front end moves something slightly physically//maybe both at the same time--likely in a major cable, and the output of the generator is lost and everything dies...because at some point it was running only on the generator because the battery wasn't in the loop sometime after the car was driven for a while warmed up. Or, as long as there is sufficient flow through a wire from the generator, the (corroded?) circuit continues, but when the engine is running slowly at stopping/idle the load is too much and it "pops" out
The car sits, front end back up, leans back to normal stance, and once the wires cool, connection is made again.
One way to test if it is the battery would be to alligator clip a 6 volt test light, with a fuse of course, directly on the battery posts and have the light up through the gap in the hood. Then, if the light goes out when the engine/power goes out, you know it is the battery. If it stays on, then you know it is electrical wiring.