It sounds to me like the rectifier might be compromised. 1 blown diode in the 4-way bridge would reduce the charge current by 1/2. Thus, everything is fine as long as nothing (head lamps, PTO, disco light ball, George Foreman electric grill) is turned ON. In that scenario the diminished charge rate is minimal, but passable until some other load is applied.
The idea of connecting another huge load (like an 8A HL bulb) to see if the same thing happens is very good, but doesn't actually quantify the problem. I'd prefer to have ammeter readings at the battery with engine running without load vs. engine running with 8A load. With a "zero center ammeter" you could actually watch the load bring the "charge rate" into the negative "discharge" zone.
Interesting that so much "jumping off" with the car is going on. Although rectifiers do give up in service, A major cause of failure is momentary reverse polarity while being "jumped off".