The website should auto navigate you to the install.php if a config.php is not found in your directory. If it is not I would venture to guess something is going very wrong and unexpected. If you could please supply some PHP error logs I would love to further help. Not sure where on the Synology Apache setup however typically these errors are sent to /var/log/apache2/error.log
The only files that need to be writable after install are the 2 log files defined by $log_file and $log_previous. During the install they are the bottom two file paths requested.