Is the wage resulting from exchanging working hours for going to treatment considered unlawful money (haram), especially since an official inauguration and commencement of work report was signed, and another report was signed to account for a period of one and a half months of work due to administrative problems, and this exchange period was approximately one and a half months?

The matter of alternating work shifts is subject to the employer's permission. If they grant permission, there is no objection. If they do not grant permission, the worker is not entitled to wages for the hours they did not work. A private employee must adhere to the contract, and their time is owned by their employer. They are not allowed to appoint someone else in their stead without permission.