Stasch, I've used one of the $5 square full-wave rectifiers for years on my Bonneville with no problems. The original ones seem to be prone to failure due to vibration. The square ones are a direct replacement. I'm looking at one now. They are encapsulated in resin and vibration-proof. I get mine at any electronics store - 35amp full wave (or bridge) rectifier. These are what Quinten refers to above as a "cubed rectifier".
The base is metal and 1" square, with a 5mm mounting hole in the middle. There are four male spade connectors sticking up out of the resin that fills the square. One connector at each corner. There is a printed marking on one side of the square base. Adjacent to the left hand connector is the mark "+", and adjacent to the right hand connector is "AC".
"+" is obviously the positive connection (earth connection to the frame or the battery on your '67 BSA). Assuming that your '67 BSA is positive earth like my '69 Triumph.
Unmarked, and diagonally opposite on the rectifier, is the "-" or negative terminal, which goes to the battery.The other two diagonally opposed terminals, one of which is already marked "AC", are connected to the alternator. It doesn't matter which way around they go.
The original rectifiers are finned for cooling, but I have never found overheating to be a problem with the square one. If it bothers you, you could put a big aluminium washer of any shape behind the rectifier as a heatsink when you mount it to the frame using a bolt through the central 5mm hole.
I hope the above is of use. Replacing the rectifier is a quick, easy, and reversible way to approach your problem. If it doesn't provide a solution you may have to consider looking at your Zener Diode. This is the device that regulates the generated voltage by shorting to earth when the voltage is over a given value (13.7 from memory).
Harder to test and not cheap to replace, the Zener is the electronic device in the middle of the deeply finned cast aluminium plate mounted at the top of your front frame tubes. In the airflow to help dissipate the heat generated when it is subjected to excess voltage. You can never tell, but I would not expect a Zener to fail in the way that you have described. I think it would be more likely to open-circuit and cause excessive voltage in the system.
This is what's inside it:https://hackaday.com/wp-content/uploads/2014/12/bridge-rectifier-shunt-before.png
The triangle thing with a line across the point is a diode, which allows electrons (i.e. current) to travel through it in one direction only.