It’s a matter of Change of Management on the project, and as such should be already part of the project plan in the first place. Every project will face issues of one kind or another, and project plan should provide how such changes are handled formally within the project.

In this case cost estimation is related to elements needed in order to achieve a final goal or deliverable objective. If budget is cut, PM needs to perform analysis of the impact and take proper action based on the result of the analysis.

It’s the customer that pays in the end, and PM just needs to make sure that they have all the data required to make right decisions on Steering Committee.