Is it permissible for an employee to deal personally with the clients of his company to make a profit from requests that the company rejects, while the employee is keen not to let the company lose the client?

If the company rejects a client's request, and the questioner's interaction with the client does not harm the company or the client, but rather benefits both parties, then there is no objection for the questioner to carry out this specific work related to orders that the company rejects. The more pious and safer approach is for the questioner to return to the company's management and explain the matter to them.