Use wget to mirror a site to a static site
It appears that the issue had to do with the fact that a "Logout" block was in the header of the main site. As a result, when wget went to pull things down, it would actually go to the logout link, and thus the rest of the files would either display a login screen or wouldn't be downloaded. By disabling the logout block OR adding --reject logout to my wget command, it seems to have fixed the issue and now the full directory structure is being downloaded. The command I ended up using was:

wget --mirror -w 2 -p --convert-links --load-cookies cookies.txt -e robots=off --reject logout https://url.org/user