Engine off, key switch off, master bat on
Bat voltage to bat on reg and at alternator - 12.02 volts

SO looks like with key off the Regulator is powered.
With key on Regulator is off - ??.

Alternator with field from battery is output 16.3 volt. Next reconnect alternator and regulator and feed reg direct from battery. That should tell the final chapter n verse. Either miswired or bad reg?

Dan Bentler