Is it permissible for an engineer to take a larger sum from his office in exchange for completing a government transaction through a third party, knowing that this is not part of his core work and that the office is not obligated to deal with this person?

It is not permissible to take a fee higher than what will be given to the actual lessee to perform the task, unless the office is clearly informed of this and agrees to the additional commission. If the office does not agree, it is not permissible to take any money from them by deceit and trickery, because that is consuming people's wealth unlawfully.