That really depends on the specifics of the network, and how the storage appliance is designed.
If you absolutely have to mount the storage directory as a filesystem, you're pretty obviously stuck using either SMB/CIFS or SSHFS (which just needs working SFTP support on the storage device). Of those two, SMB is almost always going to get better performance for anything except bulk transfers (and for bulk transfers, either one may win out). If you can get working NFSv4 support (and you absolutely should be able to if it's an Ubuntu box, regardless of what the OEM says), that will usually outperform both SMB/CIFS and SSHFS most of the time.
If you just want to throw data across the network, then your options aren't much better to be honest. SFTP and SCP are what I would probably use in your situation because regular FTP (and FTPS, which is FTP over SSL) is crap, and WebDAV (regardless of if it's over HTTP or HTTPS) was not designed for bulk storage scenarios (DAV stands for Distributed Authoring and Versioning, which pretty well describes the fact that it was designed as an alternative to FTP for pushing content to web servers). In an ideal situation for that, I would either use rsync (if I had lots of files to transfer), or possibly just netcat (if I needed to just copy one big file).