You have stated that, the certificate is not managed by you. To better assist you on this, just to clarify, is this free App Service managed certificate (ASMC) or purchased App Service Certificate (ASC) or private certificate from a third-party provider uploaded by any other user (on your subscription co-admin or any of your peer managing the certificate).
If it’s ASMC - it does not support usage as a client certificate by using certificate thumbprint.
And, you will not be able to export your App Service Managed Certificates as they are managed by the platform.
Based on your requirement, you may leverage ASC or any 3rd party certificate.
Reference doc section:
Additional info | Just to highlight:
On App Service, to access a certificate in your app code, add its thumbprint to the WEBSITE_LOAD_CERTIFICATES
app setting.
As for as converting certificate file format, if it’s 3rd party certificate, you can ask your CA to provide the certificate in the required format. There are also third-party tools that can help you convert the certificate to the proper format.