What is the ruling on paying a sum of money to a person with connections to facilitate and finalize the amendment of a contracting company's documents with government agencies, and what are the consequences for me as a partner in this matter?

If the amendments are truly your right and there is no injustice in them, then there is no harm in paying money to obtain them. Otherwise, it would be prohibited bribery. Prohibited bribery is that which is for a falsehood or injustice. However, if it is to repel injustice or obtain a right, some of the Salaf saw no harm in it.