Supervising the client's employees on an ongoing basis points away from a consulting engagement and toward ordinary employment, which the Management Consultant entry does not cover. Limited direction of a project team during a defined engagement is a different matter.
Direction of a project, not of a department
Hypothetical example: a Hampton specialist is told she will consult on a claims workflow redesign and, in the same breath, that she will manage the intake team's schedules, appraisals and hiring. The second half is the problem. Permanent supervisory authority over the client's staff is what an operational employee has, and it is difficult to reconcile with an engagement to analyse and recommend.
The workable version narrows her authority to the project: she leads a working group drawn from the department for the duration of the redesign, the department's own manager retains staffing decisions, and the engagement ends when the recommendations are delivered and implemented. That description is both accurate and consistent with the category. If the company will not accept it, the honest conclusion is that it wants an employee and the route has to change.