My '66 4-4-2 starts right up, even after sitting for weeks, but the first time that the headlights are turned on, it quits. It won't start again immediately, acts like a totally dead battery, but 15 seconds later, fires like nothing ever happened. You can then on-off the headlights without issue. Battery, battery terminals, cables, alternator, starter all good. Could a bad voltage regulator be the culprit? Or?

 

Yes VR could be going bad. I'm not sure where your VR obtains its ground - often the VR is attached to chassis metal (firewall?) to gain it's ground. Might check VR connection is clean.

 

What is the voltage at the battery with the engine running.

 

14V immediately after starting, after running a few minutes, settles at 13.1V.

 

13.1 is very weak. Thats 0.5 VDC off the surface charge on the battery.
14.0-14.2 is normal.
Grounds Grounds Grounds!!! Find 'em, clean 'em, spary 'em with battery protenant.

Do you know how to test an external VR? Don't just replace it.
Could be the diode bridge in the alternator failing.
Load test the alternator and the battery (if you're suspicious of the battery).

Oh did I mention verify all grounds?

 

Here's some great advice on how to troubleshoot alternators & regulators.
I can not the credit. One is from Harvey White the other from Joe Fisher.
Get out your DVM.

 

Thanks, and thanks for the links, too. It stayed at 14V for 5-6 minutes, then dropped to the 13.1V.

 

Heat somethings breaking down. Bridge diode.