Is it permissible to take an unrecorded leave from the company's system with the approval of the direct supervisor, knowing that the company does not compensate for unused leaves nor carries them over to the new year?

If the company's system stipulates no compensation for unused leave or its deferral to the following year, then this must be adhered to, based on the Prophet's ﷺ saying: "Muslims are bound by their conditions." The supervisor does not have the authority to act in this matter; rather, it falls to the company's management. If the supervisor denies leave due to business needs, the matter must be escalated to the company's management to consider compensation or deferring the days.