What is the ruling on taking sick leave from a private hospital for a surgical operation on a young son, if the government hospital only grants an attendance certificate in such a case?

If the system does not allow leave for a patient's escort, then it is not permissible to forge a medical certificate to obtain it, due to the forbidden deception and forgery involved. This is based on the Almighty's saying: "So avoid the uncleanness of idols and avoid false statement" [Quran 22:30], and the Prophet, peace and blessings be upon him, said: "Whoever cheats is not one of us." You can attend for some time and ask for permission when needed. A certificate of attendance from the hospital might suffice you.