Erratic Ammeter

My buddy's '63 Dodge 440 (383, 4-bbl, A/C, auto) has been having an issue. He claims that when he drives with the headlamps on the ammeter goes to full charge, drains the battery, and destroys the voltage regulator. Driving with the headlamps off, the ammeter shows just slightly to charge (which is what I would expect).

He has an old analog bench top volt meter (from Radio Shack) and made the following voltage measurement with the engine at normal operating temperature across the battery terminals measures ~16V.

We then swapped a known good battery from another vehicle and made the same measurement across the battery terminals. Again, it measured ~16V.

With the engine running (headlamps on or off) the ammeter is very rapid and erratic (see the video here ).

Where do we begin?