Is it forbidden to deal with man-made laws that regulate transactions such as companies and institutions, which most Islamic countries are based upon, knowing that these laws do not contradict Sharia and consider it a source for them?

There is no harm in dealing with laws and regulations that do not contradict Islamic Sharia, and whose ruler does not rule by other than what Allah has revealed, such as administrative laws that organize work, preserve rights, and secure people's interests. What is forbidden is that which necessitates violating Sharia or assisting in its violation.