I have bypassed security checks by overriding invoke() of the out-of-the-box WorkflowPermissionInterceptor and I am setting the task in question to editable by overriding onTaskDetailedData in task-edit-header.js and overriding onWorkflowDetailedData in workflow-form.js.
So the user, which is not initiator or assigned the task, can navigate to a document that is part of the workflow and click on the workflow link and the Workflow Details page is displayed with the task editable. Clicking Edit takes the user to the Edit task page. However, when updating due date I get the above access denied exception.
I have also setup the user as Collaborator on the bpm_package just to try that.