EDIT:!!! IGNORE THE SECOND PART BELOW
I missed the part you say "you can unplug the regulator and it still charges." You have something wrong either in alternator or a harness wiring short. Unplug the green field wire at the ALTERNATOR and see if problem still exists.
WHAT do you have for a system? IS this the older so called "single field" (grounded field) Mopar unit, or "something else?"
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EDIT IGNORE THIS FOR NOW
FIRST thing to do is confirm the battery is good or bad, or charged or not. Charge it have it tested. Then with the car running and with "it" able to show the problem, monitor battery voltage and see where it goes. If charging voltage is much above 15V you have a charging problem, likely. If voltage is below 15 most likely is a battery problem. You can also temporary swap in another good battery
MAKE CERTAIN REGULATOR IS GROUNDED. This is INCREDIBLY important. How to check: Run car, warm battery 'as normal" as you can get it, and run at fast idle to simulate "low to medium cruise" RPM
Run this test with all accessories off, and again with lights, heater, etc powered on.
Stab one probe of your meter into the battery NEG post and the other into the mounting flange of the VR. Make SURE to get through paint, rust, etc. You are hoping for a VERY low voltage, better yet ZERO, the lower the better.
If you read much of anything, above .3V (3/10 of one volt) check/ recheck regulator grounding to firewall, and ADD A GROUND CABLE between engine and firewall