I was under the impression that the FAA had determined that each of the WWII
fighters required a Letter of Authorization (LOA) to be flown by a civilian
pilot. Because of this LOA requirement, which is a practical equivalent to
a type rating, even a military pilot of that particular aircraft model could
not log PIC time as a civilian, unless he had the LOA.

But you don't need to be qualified to be PIC to log PIC, only "rated".
A "rating" is something printed on your certificate. Just like
endorsements are not required, I would think LOA's would not be.

-Robert