I have a clevo based device N14xWU slightly older than yours (march 2018) which does not exhibit any issue with the function keys.
I do however see a difference in bios vendor - where yours are INSYDE and mine American Megatrends
I assume the issue is only present when running Xorg.
Have you checked what configs has been applied with relation to keyboard?
/etc/X11/xorg.conf.d/00-keyboard.conf
I remember a topic a while ago on how special keys are default mapped.
I will try if I can look it up. I found it - not what I remembered and not of any help in this case. For the record it is Strange default configuration for F13 - F24 keys
If the system cannot register the keypress as you say showkey yields nothing - perhaps a more lowlevel issue?.
Unfortunately it is difficult to troubleshoot such lowlevel issues in a forum.
When you run showkey - you get this info, right?
# showkey
kb mode was ?UNKNOWN?
[ if you are trying this under X, it might not work
since the X server is also reading /dev/console ]