According to the PRC Nationality Law, a child born in China to at least one Chinese citizen parent is automatically a Chinese citizen. According to India's citizenship act, a child born outside India to at least one Indian citizen parent is an Indian citizen by descent if registered within one year of birth and the parent declares that the child does not hold a foreign passport. I don't know whether such a registration of the child for Indian citizenship would be considered by China as a voluntary acquisition of a foreign nationality that would cause loss of Chinese nationality.
If you register the child as an Indian citizen and get the child an Indian passport, I believe that, no matter if the child is still considered a Chinese citizen or not, the child would still get an entry/exit permit to exit China. Chinese dual-citizen children do not travel with PRC passports -- they enter and exit China with their foreign passport plus either a Chinese Travel Document (which can only be obtained abroad) or an Entry/Exit Permit (obtained in China). So even if the child is still considered a Chinese citizen after being registered as an Indian citizen, you would still be looking to get an entry/exit permit, and not a PRC passport.
I am not sure about the effects of not having a marriage certificate on getting the child an Indian passport and/or getting a Chinese entry/exit permit.