"Pulsing" voltage, RPM drop with load.

The first thing you need to do in these cases is determine the voltage drop in the ground path from the voltage regulator case to the battery negative, AND the voltage drop from the pattery POS to the VR power terminal. That is important.

This can in rare cases be a bad battery. If the battery is low, I would not pay too much attention to "key on" readings, but voltage drop. IE put one probe on battery positive, the other probe on an "ignition run" terminal such as alternator blue field wire. You want as low a reading as possible. Most drop is in the ignition switch, the switch connector, and the bulkhead connector.

Consider dutting loose the underhood "ignition run" line, us the bulkhead connector side to fire a relay, pull power off your starter relay through a breaker to the relay, and then feed the "run" buss you just cut off the relay