Is it permissible to compensate for an employee's mistake by increasing the price of the merchant's goods without his knowledge, despite the employer's refusal of that?

If an employee makes a mistake due to negligence or transgression, the shop owner is responsible for guaranteeing the errant employee, not you. If the shop owner insists on holding you responsible and you are unable to repel this injustice, then it is permissible for you to take from his money, without his knowledge, an amount equal to your right. This is known as the issue of "taking one's due" (mas'alat al-dhafar).

As for compensating your salary in the manner mentioned (buying the item for the customer), it is not permissible except under two conditions: 1. The employer's permission; because this occurs during working hours. 2. Agreement with the customer on a known amount in exchange for providing the item, or that the customer buys from you after you have owned and possessed the item, due to the Prophet's (peace be upon him) saying: "Do not sell what you do not possess."