Is it permissible to speak with the General Manager regarding his preference for certain employees in appointments without adhering to the system he applies to others?

The manager is not permitted to show favoritism to his relatives in job appointments. His duty is to act in the best interest [of the institution] and to appoint the most competent person for the job, ensuring it is done perfectly and justly, without favoritism, because that is injustice and betrayal.

If you do not fear harm, advise him to be just and to give everyone their due. Otherwise, lodge a complaint with higher authorities. Your provision will come to you, so be at ease, for "There is no creature on earth but that upon Allah is its provision," and no soul will die until it has received its full provision.