I'm an Egyptian citizen holding S2 type for medical treatment in Beijing hospital for three months.

Ahmed, yes, it is possible for you to extend it for further medical treatment, but I cannot assure if it will be easy.
You should apply for the extension 7 days prior to its expiry date. 7 working days are needed to get the result. It may cost CNY 168, but the final payment should be subject to the actual payment given by the consular officers.
It may be better to apply for an extension than changing it to another type.