When I read your post, my first thought was to recommend getting a copy of Dykes Automotive Encyclopedia. I have one and it has an amazing amount of info in easy to read, layman's terminology. Glad to see you have one as it is an old book and sometimes hard to find a copy.
It sounds like you have done a very thorough, well-thought-out process of troubleshooting the problem. I do not have a ready answer for you, but I would suggest 2 steps that might help: 1) disconnect everything but the minimum wiring for the charging and starting circuits and ignition before doing any further tests. Since our cars have simple systems with nothing much to disconnect, and you have disconnected some things already, you are probably already there; 2) check for any bare or pinched wires that may be causing a dead short which would drain the battery or cause an overload in the circuitry including the cutoff. You mentioned a restorer did some wiring work on the car so I am leaning toward the possibility that there may be a pinched or shorted wire somewhere. I would check every inch and every connector of the wiring.
Good luck with troubleshooting the problem. I would be interested in hearing what you find.
Joe- You've got a bad cutout relay. When the car is off the points should be open, otherwise you have a bridge from the battery through the cutout(points closed in contact) to the generator which has a ground. The wiring will get hot and could even start a fire. Our generators on their best day put out 20 amps. With the points closed the battery will send all it's amperage back through the circuit I described. When the battery is charging the points are closed in contact because the winding in the cutout creates a field and pulls them together( or the other way around, can't remember) Other than mounting or having "Ford" stamped on the casing what you're ordering should work or a generic cutout. Moss Motors has 7 minute tutorial on their website that explains things very clearly, type in voltage regulator or control box.