Since you're getting "Access Denied" (403 Forbidden) error, it implies there are not enough permissions to execute a pipeline in the Cloud Manager API. You probably just have "Developer" profile added in the API so you're able to fetch the pipelines. Though for the pipeline execution, you need to add at least "Deployment Manager" profile (other profiles are "Release Manager" & "Program Manager").
You can go to the Cloud Manager API in the developer console and under the API, there would be an option to Edit / Add Profiles. If you do not see those profiles listed (means you cannot execute the pipeline from the UI as well), your org admin would be able to add you to relevant profiles OR the integration has to be created by someone entitled to that profile and share the private key with you to set it up in your AIO cloud manager CLI plugin.