Well that's a new one...

Normally like you say you run into all sorts of trouble taking the old body mounts off, but in this case I guess your Bronco just wanted you to know it wasn't going to take it lying down that you got past that drama.
I'm thinking we can't rule out that it could just be a coincidence that it happened at the same time. But I hate assuming coincidences.
Of course, it could be both! Just coincidentally messed up your alternator when you did the body lift? The fact that it's doing it with the key off means it's something that is actively flailing around without a spinning alternator. But it could still be a bad diode in the alternator.
What alternator and regulator are you running? Can you feel anything going on with the regulator? If it's an old style, is it clicking? Any style is it warm or hot just sitting there?
Is there power to the Green w/red wire in the regulator's connector with the key off?
Won't hurt to pull a few fuses, just to see if it goes away, but if not then maybe disconnect the battery, then the alternator, then reconnect the battery to see what happens.
Good luck. Definitely report what you've done and found. This isn't the normal body lift discussion that's for sure!
Paul