TeleMessage supports the display of WhatsApp Screenname/Nickname for both WhatsApp Cloud and WhatsApp Phone Archiver products.
You can see the difference between a nickname and a local address book name in the image below.
Note that on both platforms, the screen name will be visible only after the contact sent you a reply.
So it won’t be visible if you are making the first few WhatsApp interactions with him/her.
Only after the other party responded to you over WhatsApp, will you start seeing the screen name.