According to the PMBOK Guide, 6th edition, section 13.1.3.1, the stakeholder register is a project document that identifies the stakeholders who have an interest in the project or may be affected by the project. The stakeholder register should be updated whenever there is a change in the stakeholder community, such as when a new stakeholder is identified or an existing stakeholder leaves the project. The new business unit manager is a new stakeholder who has authority and influence over the project, and therefore should be added to the stakeholder register. The stakeholder engagement plan, which is a subsidiary plan of the project management plan, describes the strategies and actions to engage the stakeholders and manage their expectations throughout the project life cycle. The stakeholder engagement plan should also be updated to reflect the new stakeholder's current and desired level of engagement, as well as the communication and engagement activities planned for the new stakeholder. The other options are not the best next steps for the project manager, because they do not address the need to formally identify and document the new stakeholder and plan for their engagement.
Contacting the previous business unit manager may be helpful, but it is not a formal project document.
Sending the new manager a copy of the project charter may be informative, but it does not capture the new stakeholder's needs and expectations. Scheduling a meeting with the new manager may be beneficial, but it should be done after updating the stakeholder register and the stakeholder engagement plan. References:
* PMBOK Guide, 6th edition, section 13.1.3.1, page 513
* PMI-PBA Exam Practice Test and Study Guide, 2nd edition, chapter 2, page 30