Is accepting private work from a mediator who dealt with a client of the company the questioner works for considered forbidden?

The contract is the law between the contracting parties, unless it violates a Shariah ruling, and adherence to it is obligatory. Since the agreement was made that you would not deal with your company's clients for your own private account, you must fulfill that, and it is not permissible for you to deal with them, neither during working hours nor outside of them. As for those other than the company's clients, there is no harm in dealing with them, but outside of working hours.