The project manager is responsible for ensuring that the project's lessons learned are captured, documented, and stored in the lessons learned register. The lessons learned register is a project document that records the knowledge gained from the project activities and processes, and that can be used in the current project and entered into the lessons learned repository for future projects12. The project manager should take ownership of documenting the occurrence that the SME identified, as it is part of the project manager's role to facilitate the lessons learned process and to encourage the project team and other stakeholders to share their experiences and feedback. The project manager should also verify the accuracy and completeness of the information, and ensure that it is relevant and useful for the project and the organization. The project manager should update the lessons learned register throughout the project, not only at the end, as this allows the project team to learn and improve continuously. This is part of the Manage Project Knowledge process, which aims to use existing knowledge and create new knowledge to achieve the project objectives and contribute to organizational learning13. The other options are not the best choices because they do not reflect the project manager's accountability and leadership for the lessons learned process. Assigning the documenting task to a project team member (option A) may delegate the project manager's responsibility and reduce the project manager's involvement and oversight of the lessons learned process. Asking the SME to document and capture the occurrence (option B) may shift the burden to the SME and discourage the SME from sharing more information and insights. Requesting the SME to wait until the project closure phase (option C) may delay the documentation and dissemination of the valuable knowledge and prevent the project team from applying it in the current project. References:
Project Management Professional (PMP) Reference Materials:
A Guide to the Project Management Body of Knowledge (PMBOK Guide), Sixth Edition, Chapter 4: Project Integration Management, pp. 71-123 Lessons learned - Project Management Institute Project Knowledge Management According to the PMBOK Professional in Business Analysis (PBA) Reference Materials:
Business Analysis for Practitioners: A Practice Guide, Chapter 6: Traceability and Monitoring, pp. 97-98 Lessons learned - Project Management Institute

According to the PMBOK Guide, the project manager is responsible for managing the project team and resolving any conflicts that may arise among team members. The project manager should use appropriate conflict resolution techniques, such as problem-solving, compromising, or collaborating, to identify the root causes of the issues and guide the team members toward a mutually acceptable solution. Removing one of the team members, warning them about their behavior, or training them on communication techniques and interpersonal behavior are not effective ways of handling this situation, as they do not address the underlying problems or foster a positive team environment. References: PMBOK Guide, 7th edition, page 223-224; PMBOK Guide, 6th edition, page 349-350.

When a key stakeholder introduces a new requirement, it is essential for the project manager to request further clarification. This ensures that the requirement is fully understood and its impact on the project schedule can be accurately assessed. The PMBOK Guide emphasizes the importance of clear communication and understanding stakeholders' needs for successful project management1. Additionally, the PMI-PBA highlights the need for detailed requirement analysis to manage changes effectively and align them with business objectives2. References:
* Stakeholder Analysis: Pivotal Practice to Project Success | PMI1
* PMI Professional in Business Analysis (PMI-PBA) Handbook2

In a scenario where a company is not yet profitable and is facing project delays with an overwhelmed staff, the project manager should prioritize the projects based on their potential to increase profit. This is in line with the principles of project financial management, which suggest that prioritizing projects that align with the company's strategic goals, such as financial objectives, is crucial. By focusing on projects that are more likely to contribute to the company's profitability, the project manager can help ensure that resources are allocated effectively and that the company's financial position is strengthened. This approach is supported by the PMBOK Guide, which emphasizes the importance of aligning projects with organizational strategy and business objectives.
References:
PMBOK Guide - Sixth Edition
PMP Examination Content Outline - June 20191
Professional in Business Analysis Reference Materials2
Project Prioritization Process: 5 Strategies to Prioritize Your Projects3 Analytic Hierarchy Process Prioritize Projects | PMI4

According to the Professional in Business Analysis (PMI-PBA) Examination Content Outline1, one of the tasks of a business analyst is to "communicate and manage solution scope boundaries and approval processes to stakeholders" (Task 5.2). This implies that any changes to the solution scope should be approved by the relevant stakeholders and follow a predefined process. The PMBOK Guide2 also states that "the project manager, project team, and other stakeholders are responsible for determining how the project scope will be managed, how scope changes will be integrated into the project, and who will approve the scope changes" (Section 5.6.1). Therefore, the project manager should advise the team not to engage in modifications unless it follows the proper change approval process, as this would ensure that the scope is controlled and aligned with the project management plan. References:
* 1: Professional in Business Analysis (PMI-PBA) Examination Content Outline
* 2: A Guide to the Project Management Body of Knowledge (PMBOK Guide) - Sixth Edition