The project manager should inform the sponsor and document the request in the stakeholder engagement plan, as this is the best way to handle the situation without compromising the project scope, budget, or schedule.
The sponsor can then decide how to deal with the key stakeholder's request, and the project manager can update the stakeholder engagement plan accordingly. This will also help the project manager to maintain a positive relationship with the key stakeholder and avoid any potential conflicts or misunderstandings.

According to the PMI-PBA Exam Practice Test and Study Guide, 2nd edition, chapter 8, page 179, incremental budgeting is a technique that allows agile teams to estimate and allocate funds for each iteration or release of a product. This technique enables the teams to respond to changing requirements and priorities, and to deliver value to the customer faster. Incremental budgeting also reduces the risk of over- or under- budgeting, as the teams can adjust their estimates based on the actual costs and benefits of each increment.
The other options are not suitable for agile teams, as they do not support the agile principles of flexibility, collaboration, and customer satisfaction. Requesting the finance team to allocate a budget based on historical data may not reflect the current or future needs of the product. Providing the budget to the finance team for only the planned increments may not account for the changes that may occur during the product development.
Extrapolating the budget based on the current cost performance index (CPI) may not be accurate or reliable, as the CPI may vary depending on the scope, schedule, and quality of each increment. References:
PMI-PBA Exam Practice Test and Study Guide, 2nd edition, chapter 8, page 179 PMBOK Guide, 7th edition, section 8.1.3, page 269

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

According to the PMBOK Guide's Change Management Process, when a new change is suggested, it should first be identified as a change request by any stakeholder. The project manager and/or project team members then analyze the impact of the change request on the project. This includes documenting the change request and presenting it to the Change Control Board (CCB) for review and decision1. Accepting a suggestion without a formal change request could bypass necessary analysis and documentation processes that ensure the change is beneficial and does not introduce unforeseen risks or issues. References: PMBOK Guide's Change Management Process1; Professional in Business Analysis Reference Materials2.

According to the PMBOK Guide, 6th edition, section 4.7, one of the main outputs of the close project or phase process is the final product, service, or result transition. This involves transferring the ownership and responsibility of the project deliverables to the customer or sponsor, and obtaining their formal acceptance and sign-off. The project manager should ensure that the project scope and quality criteria are met, and that the deliverables are aligned with the project objectives and stakeholder expectations. The other options are not appropriate for the end of the project, as they are either part of the planning or monitoring and controlling processes, and do not involve the client's verification and acceptance. References: PMBOK Guide, 6th edition, section 4.7; PMP Practice Exam, question 86.