No issues if your wife and you are in different sponsorship. When you have the new born , your wife will be the sponsor/ guardian for baby's visa?
The Visa provided will be a 6 Month Visit Visa.
If spouse and your company doesnt not provide visa for our baby, there is problem...baby cannot stay. You have to complete the procedure of residency within the period of Visa (3Months or 6 Months...whatever is the visa duration)...

MAke sure your wife is registered at Hamad hospital prior to delivery even if you are showing her in a private hospital.