I’d suggest checking the “earth” straps between battery and chassis and chassis and engine with considerable care as this is usually the cause of odd (and in your case extremely odd) electrical problems. A quick diagnosis might be achieved by connecting the alternator casing directly to earth using a jump lead. And also check the live battery lead. It sounds to me as though a high or perhaps a slightly intermittent resistance in one of the main feeds to or from the battery may be behind the issue.
Why, oh why, did people used to put the engine "earth" lead in the most inaccessible possible place?
Also be aware that an alternator seems to be capable of overcoming a high resistance in the battery charging circuit, which of course includes all the above straps and leads, by winding its output voltage up, notwithstanding any internal or external regulation. But the end result is that it both overcharges the battery (one can’t actually get a battery voltage beyond a certain point, but I hope you see what I mean) and is perfectly capable of burning its internal insulation out. I don’t claim to fully understand the process but have experienced it - I went through three alternators rebuilds before I got to the root of the problem.
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