The solution as it turns out is to reset the file permissions on the core files.
Provided you have SSH access to your server, the following commands may fix your issue.
Reset the permissions of all files to 664:
find /path/to/site/ -type f -exec chmod 664 {} \;
Reset permissions of directories to 775:
find /path/to/site/ -type d -exec chmod 775 {} \;
Reset the group to the wordpress group (or whatever group makes sense for you)
chgrp -R wordpress /path/to/site/
After running those commands I was able to successfully update WordPress with no further issues.