The Zoomboni has a dead alternator.

Ok, one last try.  Key on. Engine off

Measure the voltage of everything going into the alternator and the voltage regulator.  Report those, we'll go from there.

So new alternator.  New pigtails,  new starter solenoid.  New battery.

Still not charging.  berkeley this berkeleying E36 M3!

Too soon to ask how much?

How is the ground from the engine to the frame?  Without that it can cause charging issues.

Also, is the fusible link in the yellow/white wire good?

Hang on.  Pause.  Wait.  Chill.    By all means, stop throwing parts at it, and don't take it to a shop for diagnosis.  Shops are like doctors.  You show up with a headache and they say take an aspirin, that will be $120.  In your case, you know you have a tumor on the side of your head and you've tried aspirin and it didn't work.  A shop will say "bum alternator" and charge you for parts and labor that won't fix the problem.  Shops deal with cookie cutter problems on stock vehicles.  They won't dig deep unless you get super lucky and the tech is a Ford wizard and has 6 custom Mustang projects of his own.  Even then, it might take him a while to get caught up on your learning curve.

From one of those pigtails there should be a green wire with a red stripe.

That one is your dash indicator lamp wire.  Since you don't have a dash light, my guess is that wire doesn't go anywhere, right?  Without it, the alternator won't charge.  Let me know where that wire goes and I'll tell you how to fix it if that's the problem.

Another dumb thing to check. It's really not that dumb, because I see it all the time.  The wire from the alternator to the starter solenoid:  Is it on the same post as the wire going to the battery, or on the post with the wire going to the starter?  It needs to be on the battery side.  Otherwise, it would only charge during cranking, not during running.