My wife and I both have valid RPs for Qatar. I just wanted to make sure I have all the right documents so that the little one can get her RP stamped once we are in Qatar.

The only difference I see from the process you have noted (info from the US Embassy) is that I will need to get the birth certificate notarized by the secretary of state where the child was born, then notarized by the State Department in DC and then the US embassy in Qatar. And once all this is done, I will have to take the documents to the Consular Affairs Section of Qatar's Ministry of Foreign Affairs to authenticate the Embassy's seal and signature. (No role for the Qatar Embassy in US)

From my research I found out that all these steps are necessary because Qatar is not a Hague Convention country. Otherwise an Apostille would have sufficed.

Good luck with your Visa Stamping!