To move a part of the Mac OS X system to an external drive, that is not permanent
to the computer itself & does not contain the rest of the OS X, is asking for trouble.
Before attempting to modify or change how the system works, you should backup
the entire content and not just by Time Machine file methods. You may need to
restore the entire content, and without a disk image or bootable clone copy of that
system as you have it, would not be as easy or directly possible to restore.
Some applications that save their copies to the system's document folder would
not likely find it on an external drive. Or when it fails to be identified correctly,
you may lose direct access to the items sent there.
I've seldom relied on the system's Documents folder, but instead have created files
for my own content in locations where I hope to retrieve them. Things happen, so
the backup and a backup of the other backup, is recommended even if you don't
try moving system folders to another drive, an external archive that may be lost.
Good luck & happy computing! 🙂