I used this one - there is a free version but for $0.99 I chose to support the dev.

https://play.google.com/store/apps/details?id=com.theolivetree.ftpserverpro&hl=en

Works awesome. You can find setup instructions here:

http://nvidiashieldzone.com/shield-android-tv/tutorial-ftp-server-on-shield-android-tv/

If you do go with just adding an external drive and consider yourself technical, I'd install an FTP server on your shield
The one I've used before is Ftp Server Pro . With that turned on you can use something like Filezilla to copy your media over to the external using your computer. You'd have to install Sideload Launcher or similar so you can access the Ftp Server Pro on your shield.

External SD card should not be an issue. You can select the path of the server and access and manage via a client.

Personally, I use ES file explorer 3.2.5.5 (an older version). They say the current version is compromised for the original dev sold it to the Chinese.

Check out "Ftp Server Pro"

https://play.google.com/store/apps/details?id=com.theolivetree.ftpserverpro

Or

Check out "Ssh Server Pro"

https://play.google.com/store/apps/details?id=com.theolivetree.sshserverpro

I've been using this one for a while to transfer things between my phone and my tablet.

https://play.google.com/store/apps/details?id=com.theolivetree.ftpserverpro