When faced with a delay in a critical path activity, it is prudent for the project manager to thoroughly evaluate the proposed fallback plan, considering its impact on the project's scope, time, cost, and quality. If the fallback plan is viable and justifies the additional cost, the project manager should proceed with submitting a change request. This process involves documenting the proposed changes, assessing the benefits and risks, and seeking approval from the project sponsor or change control board. The change request should detail the reasons for the deviation from the original plan, the expected outcomes, and the justification for the additional expenditure. This approach ensures that any alterations to the project are well-considered, transparent, and formally approved, maintaining the integrity of the project management process123.
References: (Project Management Institute. (2017). A Guide to the Project Management Body of Knowledge (PMBOK Guide) - Sixth Edition. Newtown Square, PA: Author; Project Management Institute. (2013).
Practice Standard for Project Risk Management. Newtown Square, PA: Author; Project Management Institute. (2019). PMI Standard for Project Management. Newtown Square, PA: Author).

The first thing the project manager should do is to evaluate the impact that the inclusion of this new requirement will have on project performance. This is because the project manager needs to assess how the new requirement will affect the project scope, schedule, cost, quality, risk, and stakeholder satisfaction. The project manager should also consider the feasibility, desirability, and viability of the new requirement, and whether it aligns with the project objectives and business case. Based on the impact analysis, the project manager can then decide whether to accept or reject the change request, or to escalate it to the appropriate authority for approval. The project manager should also communicate the results of the impact analysis to the stakeholder and explain the rationale behind the decision. References:
* Change Management | What is Change Management | Prosci
* What Is Project Scope? Scope Management Steps, Tips & Tools
* Definition of Change Management - Prosci

During execution, any addition to scope must go through the formal change control process. The project manager should initiate a change request for proper evaluation and approval. This ensures that scope, budget, and schedule impacts are assessed before proceeding.
According to the PMBOK Guide - Seventh Edition, any changes to the baseline during execution must follow the change control procedures defined in the governance structure.
Option A risks scope creep.
Option B is helpful but not sufficient as a next step.
Option C closes the door without considering the critical nature of the request.
References:
PMBOK Guide - Seventh Edition, Delivery and Planning Performance Domains PMI Change Control and Governance Guidelines

According to the PMBOK Guide, a stakeholder engagement plan is a document that identifies the current and desired levels of stakeholder engagement, and the strategies and actions to increase or maintain stakeholder support throughout the project. A stakeholder engagement plan is an essential tool for managing stakeholder expectations, addressing stakeholder concerns, and resolving stakeholder conflicts. Conflicts between project stakeholders can result in delays, rework, scope changes, or even project failure. Therefore, the project manager should focus on revisiting the stakeholder engagement plan, focusing on specific overlooked stakeholders, to identify the root causes of the conflicts and the best ways to engage them. By doing so, the project manager can improve the stakeholder relationships, align the stakeholder interests, and facilitate the deliverable and output approval process. References:
* PMBOK Guide, 7th edition, Chapter 8: Communications, Section 8.3: Manage Communications, pp. 126-1271
* PMBOK Guide, 7th edition, Chapter 9: Stakeholders, Section 9.2: Plan Stakeholder Engagement, pp. 141-1431
* PMBOK Guide, 7th edition, Chapter 9: Stakeholders, Section 9.3: Engage Stakeholders, pp. 144-1451
* Professional in Business Analysis (PMI-PBA) Handbook, Section 5: Exam Content Outline, Domain 2:
Planning, Task 2.4: Develop a stakeholder engagement plan, pp. 22-232

According to the PMBOK Guide, 6th edition, section 11.6.1.1, the project manager is responsible for monitoring and controlling the project risks and ensuring that the project objectives are met. The project manager should also identify and analyze any new or emerging risks that may affect the project performance or scope. In this scenario, the project manager should first work with the team and others in the network to assess and prioritize the obstacles. This will help to determine the impact and probability of each obstacle and decide on the best course of action to address them. The other options are either premature or inappropriate, as they do not address the root cause of the problem or respect the authority and responsibility of the project manager. References: PMBOK Guide, 6th edition, section 11.6.1.1, page 442.