I started scraping away some grime from the transfer case (you should see the pile of crud on the floor underneath!), and there was a nice S/N plate in very good condition:  S/N 260420.  The transmission is casting T-90A-1, with date 2-23-49 (date applied upside down).

There are some more casting and assembly dates to look for that I haven't found yet.  So far, everything on the Jeep (other than the body and engine, which are known not to be original) is consistent with the serial number (23833) on the registration. (According to the best Jeep historian on the web.)  This number is on the registration papers from the last two owners before me, dating back to 1989, when the people I bought it from bought it from the owner before them.  The original body tag is missing (presumably it went with the original body), so this is the best we've got.

The S/N is from 1949, but the Jeep was registered as 1950.  I'm told this is not unusual; at the time, the registration date could have been the date when it was first registered/licensed.