Would it work for you to also exclude the symbolic links and the directory?
rsync -vzrP --delete --exclude '.git' --exclude 'awstats-icon' --exclude 'awstatsicons' --exclude 'icon' ~/Sites/path/to/sitename/ user@server.com:/path/to/sitename/
This would prevent pushing the symbolic links and directory, while preserving (not deleting) the remote symbolic links or directories with the same names.