data source issue in service

There is something that doesn't match. Let's recap.

You have a connection to a RestAPI with Power Bi Desktop that has an authentication inside the code because you are saying you are specifing Anonymous Credentials for the source permission.

Then you are talking about Power Bi Service with an error message of data source credentials and specifing that you are providing those like using a different  source permission rather than Anonymous like with desktop.

Be sure that you are doing the exact same thing with both of them. The error message is clear, the credentials provided can't connect the source.

I hope that helps,