When a customer requests additional functionality that risks the delivery date and conflicts with the agreed scope, the project manager should engage with the team to assess the effort required and then raise a change request. This process involves analyzing the impact of the change on project constraints and documenting the change request for further review by the Change Control Board (CCB). It is a structured approach that ensures any alterations to the project scope are carefully considered and approved before implementation, thus maintaining control over project outcomes.
References: (PMBOK Guide's Change Management Process For PMP Certification1, Professional in Business Analysis Reference Materials | PMI2)