TL;DR
You lack clarity about your role, the role of the project manager, and the necessity for change control on a formal project. It is extremely likely that you lack the delegated authority to make resource allocation decisions for the project, and that attempting to set engineering priorities directly would bypass the project's change control process. You should clarify your role and the limits of your authority with the functional manager that you report to.
The Roles of Design and Controlled Delivery
The UX designer however is responsible for the UX vision.
No. From a project perspective, the UX designer is a resource responsible for delivering user interface design tasks related to the project's goals.
The vision for the product is ultimately the responsibility of the stakeholders, although it's quite likely that they look to you—perhaps even rely on you—for subject-matter expertise. However, unless you have been explicitly granted the authority to unilaterally define the interface without an approval, oversight, or budgeting process, it seems likely that the organization has not made the limits of your role clear to you.
The PM returned an email saying "Please [don't] do any changes to the GUI without PM approval".
The UX designer is not the person who sets project priorities or allocates budgetary or engineering resources. While recommending design improvements is part of your job, deciding whether the project should expend resources on implementing the design changes is the job of the stakeholders, who then implement their decisions through the chain of command for the project. In your case, that appears to be the Project Manager.
In addition, the Project Manager is generally the person responsible for enforcing the project's change control process. Change on a project generally must be controlled, and on a large project it must be formally controlled. Bypassing the change control process is not only stepping on the political toes of the Project Manager, but it may also put the project itself at risk in a variety of ways.
Integrate with the Process
User interface design should be part of a project, not outside it or above it. Therefore, you should be following the same change control process as the rest of your project team. It's probably a good idea to talk to both the project manager and your functional manager to identify the project's core processes, and to delineate the roles and responsibilities your organization has defined for your position.
That's often all it takes to clear up misunderstandings. If it is not a misunderstanding, but rather a role that you feel limits you in ways you can't tolerate, you have some hard career choices to make.
