Parents under different sponsors, can father be newborn's sponsor?

Asssalamualaikum and Hi. Me and my wife are working and are under different sponsors. In the event of having a baby in Saudi Arabia, is it possible to get the newborn child under the father's sponsorship? If yes, are the following documents good enough to get the iqama process done or will I require anything else?
a)Birth Certificate of Child
b)Passport of the Child
c)Indian Marriage certificate of parents(attested by Saudi Embassy in India)

It is enough !
And it is very common case of your situation for all families here.

And almost all children being born here or even outside KSA and want to bring them would be under their father's sponsorship.

But there are exceptions according to parents advantage and their choice like if mother's work privileges are better from her husband's work.. like insurance, tuition allowances,..etc.

Congratulation for your new baby! :)