Please note that when using the Generic SIP Trunk template, it is configured to read the Caller's Name from the 'From: UserPart':
<field name="ParameterIn" custom="" parameter="FromUserPart">$CallerName</field>
Since you are not using a 3CX supported provider, you should contact your provider to confirm which specific SIP field they use to send the CNAM to the PBX.
Once confirmed, navigate to Admin > Voice & Chat and locate your provider. Click the three dots icon on the right of the entry and select 'Export' to download the template. This is the file that needs to be modified.
After making the necessary changes, delete the existing trunk and import the modified template back into the PBX to create a new trunk. You can do this navigating to Admin options > Advanced > Templates > Provider Templates > Import Provider.