I finally got in contact with a super Box Admin/Guru in our organization. Here's what I learned about shared folders to make rclone or ftp/ftps work properly.
- The sharing party needs to share with one of the following permissions: Editor, Viewer Uploader, Previewer Uploader, or Uploader.
- I think they need to explicitly list you as a collaborator. Then you get a notification in your account and the shared folder is visible in your Box account's file tree.
And then I can at least use FTPS to access the shared folder... if FTPS can do it, so too can rclone. I will probably use rclone for this very large transfer!
So it's a permission error and a newbie error on my part. So I will click this as "solved." I guess Box is not as bad as I thought it was!
EDIT: If you use FTPS and your organization uses some kind of smart card authentication for the Box service, you may need to visit your account settings and create a password (something I wasn't aware of).