For a moment I thought I had this sorted. I had slightly different versions of Django between my development and production environments, and some of the static files for the admin pages had changed. It made sense that when Nginx redirected the static file requests, the new resources weren't there. Upgraded both environments to the latest Django (now 1.10.1), recollected static files, and got it to work.
Just to prove the point, I logged out, and tried to login with a different account, but got the 405 page again. So I tried the first account again with no success. Except for that one successful login, everything continues to return the 405 error.
From the errors in the Django log, it still seems like I may be missing a static file, or maybe a permissions problem on a directory, but no clear pointer to a specific file or directory.
Anyone have any tips on how to troubleshoot this?
Here is the specific error logged by Django's debug logger: