What is the ruling on paying a sum of money to one of the engineers of the main company in exchange for expediting the completion of our documents, knowing that he does so outside official working hours and using the company's official system?

If preparing documents is part of the engineer's work at his company, then it is not permissible for him to take a fee for it except with the company's permission, even if it is done outside official working hours. Otherwise, it would be considered among the gifts to workers, which are unlawful gains (ghulul).