On the server, I had to remove the symlink to the content folder since Dropbox no longer follows the contents of symlinks and implemented the changes above to index.php instead.
However, a few other things needed to be done with the permissions in order to get the panel to work on the server, in addition to having local changes within the shared Dropbox folder successfully sync. I figured others might run into this issue so here’s what I did to get it to work:
Install and link a new Dropbox account on Linux
Change the permissions and ownership of the Dropbox folder to match the Kirby installation so that Kirby can both read and write to it: sudo chown -R user:www-data /home/user/Dropbox/ sudo chmod -R 775 /home/user/Dropbox/