Hello MrSilver91,
Good day! Hope this finds you well.
Based on your description, you can create a custom caller ID policy in the Teams Admin center. For the outbound calls, the telephone number assigned to the user, which is the default shows. You can replace this with the Calling Party Name or CNAM set on the outbound PSTN call.
More info here: Manage caller ID for users - Microsoft Teams | Microsoft Learn
However, the CNAM information for the call may or may not be routed with the call because as this information depends on how the country/region has implemented CNAM (if at all). The reliability of CNAM delivery with the call varies depending on the country/region and carriers that handle the call--either as an intermediary or a terminating carrier.
Reference: More about Calling Line ID and Calling Party Name - Microsoft Teams | Microsoft Learn
Best Regards,
Ashraf.