According to the Agile Methodology1, the product owner is responsible for defining the product vision, managing the backlog, and ensuring that the development team is working on the right priorities. The product owner is also the primary customer advocate and the link between the business and technology strategy.
Therefore, the product owner's engagement and interest are crucial for the success of the project. If the product owner changes during the project, the project manager should reevaluate the new product owner's needs, expectations, and communication preferences, and adjust the collaboration accordingly. This can help to avoid delays, misunderstandings, and conflicts, and ensure that the product owner is aligned with the project goals and the development team. The project manager should also facilitate the transition of the product owner role and ensure that the new product owner is familiar with the product vision, the backlog, and the agile principles and practices. References:
* 3: Agile Methodology: Role of Product Owners & Development Teams

The project manager should prioritize the product backlog looking for high business value and low effort, and adjust the project budget and staffing to account for those items, as this is a strategy that aligns with the agile principles of delivering value early and often, and responding to change over following a plan1. By focusing on the most valuable and feasible features, the project manager can maximize the return on investment and customer satisfaction, while minimizing the waste and risk. The project manager should also collaborate with the product owner, the team, and the stakeholders to review and reprioritize the product backlog regularly, and to adapt to the changing requirements and constraints. Reducing the project staff and adjusting the scope and timelines accordingly in support of the budget reduction (option A) may not be a viable strategy, as it could compromise the quality and functionality of the product, and create dissatisfaction and demotivation among the team and the customers. Balancing the project team in order to accommodate the budget challenges, and providing training to mitigate the risk of low performance (option B) may not be a feasible strategy, as it could incur additional costs and time, and disrupt the team dynamics and productivity.
Restricting the project scope only to basic critical features and running the project with the current staffing level until all of the budget is consumed (option D) may not be an effective strategy, as it could ignore the feedback and expectations of the customers, and miss the opportunities to deliver incremental value and improve the product. References:
* 1: Agile Practice Guide, Project Management Institute, Inc., 2017, p. 8.

Inviting a design team member to the next daily meeting facilitates direct communication and collaboration, which is in line with Agile principles. This approach promotes quick resolution of issues and supports the iterative nature of Agile project management, where cross-functional team collaboration is key to delivering value in each iteration1.
= The iterative project management process emphasizes the importance of collaboration and transparency to address pain points and avoid stagnation of progress1. Additionally, the issue log is a tool for tracking and managing issues, but it does not replace the need for direct communication and immediate action when possible2

The best way to resolve any stakeholder issues is to communicate with them directly and openly. Setting up a meeting with the stakeholder will allow the project manager to understand their expectations, needs, and concerns, and to negotiate a mutually acceptable solution. Moving to a predictive approach, talking to the team separately, or scheduling a team retrospective will not address the root cause of the problem, which is the stakeholder's dissatisfaction with the product concept. Moreover, these options may create more confusion, conflict, or delay in the project. References:
* PMBOKGuide, 6th edition, page 518: "The project manager should proactively and regularly engage with stakeholders to understand their needs and expectations, address issues as they occur, manage conflicting interests, and foster appropriate stakeholder engagement in project decisions and activities."
* PMI-PBA Guide, 1st edition, page 35: "The business analyst should communicate with stakeholders to elicit, analyze, and validate their needs and expectations, and to facilitate collaboration and consensus-building."

Checking the risk register for a prepared response to a resource being suddenly unavailable is a proactive action that the project manager should take to deal with the situation. The risk register is a document that records the identified risks, their causes, impacts, probabilities, and responses. By consulting the risk register, the project manager can see if there is a contingency plan or a fallback plan for the loss of a key resource, and implement it accordingly. Checking if the team member's activities are sufficiently documented to facilitate handover is another important action that the project manager should take to ensure the continuity and quality of the project work. The team member's activities may include tasks, deliverables, dependencies, assumptions, constraints, and other information that are relevant to the project scope, schedule, and cost. By reviewing the documentation, the project manager can identify the gaps, risks, and issues that may arise from the absence of the team member, and take appropriate measures to address them. References: (Professional in Business Analysis Reference Materials source and documents)
* PMBOK Guide, Seventh Edition, Section 11.4.3, "Risk Register"
* PMBOK Guide, Seventh Edition, Section 4.3.3, "Project Management Plan"
* Professional in Business Analysis (PMI-PBA) Guide, Section 4.2.2, "Documentation"