This is not possible to be done on the phone.
If you change the caller ID through the admin portal, the phone needs to reboot for the change to take effect.

You might be able to pull it off you have a cordless yealink with multiple handsets. And if you have spare licenses. You could then assign a different license to each handset and set up unique caller ID for each handset. I would advise against this, mainly because you would be paying double to have 2 licenses on 1 phone.

You can consider using 1 of the apps.
- desktop app (that you can download on a computer)
- browser app (no need to download)
- mobile app
Then you could set 1 caller ID for the app and another for the deskphone. It is also easy to change the caller ID on the app. No need for reboots, restarts or anything like this.