So in El Capitan (and probably earlier), I was able to go to Library/Application Support/MobileSync/Backup and copy that folder to an external hard drive, delete the old folder on my internal drive, and use terminal to create an alias folder of the one on my external hard drive to place where the folder went on my internal drive (
ln -s /Volumes/INSERT HARD DRIVE NAME HERE/MobileSync/Backup ~/Library/Application\ Support/MobileSync/Backup). The alias folder had the same name ("Backup") as the "real" one and so, as long as I had my external drive connected while I was trying to run iOS device backups, it worked. iOS device backups were saved to the external hard drive instead of internal SSD.
However, in Sierra, this doesn't work. Even if I do it the other way of highlighting the "Backup" folder on the external drive, pressing ⌘-L, and manually moving the alias folder to where it would go on my internal drive, it still doesn't work. Everything in Finder looks just like it did in El Capitan, however I get "iTunes could not backup the iPhone because an error occurred" when I try to backup my iPhone.
I am running the latest version of Sierra, the latest version of iTunes, and my iPhone is running the latest version of iOS. I've restarted both my Mac and iPhone and even tried backing-up a family member's iPhone, but to no avail.