Is the additional amount of money requested for my mother's Hajj visa, which is paid to an employee at the Saudi Embassy in exchange for arranging the visa, considered a bribe or a fee for a service?

If the money is paid to the employee and does not go to the embassy's budget, then it is a bribe and a forbidden gift, and the employee takes it as unlawful money. However, if you cannot obtain the visa except by paying this amount, then there is no blame upon you, and the sin is upon him. Jurists have made an exception from the forbidden bribe for that which is paid to obtain a right, so it is forbidden for the recipient but not for the giver. This is supported by the statements of Sheikh al-Islam Ibn Taymiyyah and al-Taqi al-Subki regarding the permissibility of paying it if it is to obtain a right or to repel injustice, while it remains forbidden for the recipient.