Saving macOS Catalina Screen Recordings Cache to External Drive
You can try the following option:

$ cd ~/Library/Containers/
$ mv com.apple.QuickTimePlayerX/ com.apple.QuickTimePlayerX.backup/
$ ln -s /Volumes/&lt;NAME_OF_VOLUME&gt;/com.apple.QuickTimePlayerX/ .


If the above option didn't help, then I would suggest you to:

Check the first comment on this page shared by bmike.