I suppose you are still in your country.. based on my family's experience, my wife came here first and after a few months she took me and our child thru visit visa which is valid for 6 months.Then after coming here, she applied thru the Labor Department for residence visa for our daughter and thank God she got it.No problem with Hamad Hospital, they will give you NOC (No Objection Certificate) for taking the residence visa for your baby.

Hope this helps..