We have the current situation:
As starting point we used https://learn.microsoft.com/en-us/azure/api-management/api-management-howto-protect-backend-with-aad, but most of the the explanations we found concerned using AD, which we don't need as far as we understand.
We tried to implement the following OAuth 2.0 Postman Authorization configuration into APIM (which actually works in Postman).
Is there a simple and straight forward way to tell APIM to do a call to the token URL with a given ClientId and secret and add the authorization header with a bearer token to the backend API?
Yes - you can do this and here is a Curity resource that follows a similar process:
Your case is a little different but uses the same building blocks. You just need to adapt the OAuth message to use the Client Credentials flow.