Visa extension in India for a baby U.S citizen

I'm an Indian citizen living in the United States on a dependent visa. My 5 month old son, was born in the U.S and now we are in India.
On arrival, the stamp in the baby's passport says something like, the stay should not exceed 180 days. (Until May 2020).

I have applied for a Visa extension online, for the baby, and received a certificate in the email (without any fee or personal visit to the office).
It says -

Service Granted - Registration Certificate/Residential Permit Valid until October 2023

I just wanted to know, if we wish to extend our stay even after 180 days (May 2020), is that RC is all that is needed to be produced at the airport when returning to the U.S.
Do we need any other certificates to be produced for the baby in India or the U.S.