No Last Name & Child Birth In USA

My son was born in Nov, 2015 in United states. And as per procedure, hospital was supposed to take care of filling for my son's SSN and birth certificate. While filling up the forms they wanted mother's first and last name and in my case I have only first name in my Indian Passport and FNU(First name) in my visa. Check the visa story here.

My name on Indian Passport: XYZ
My name on USA Visa: FNU XYZ

Birth certificate and passport are the documents which are going to be with my son for life long and by no means I wanted my name to be printed as FNU in my any of my son's document. It has several risks and pain associated to it as well:
1. While traveling to any other country other than USA, how am I gonna proof that I am the Mother of son, as my real name is XYZ but my son's passport and birth certificate will show FNU XYZ.
2. While getting admission to schools and colleges, there would be mother's name mismatch.
3. While making any future investment for the kid.
4. While buying any property or for any other legal formalities that require mother's name.
5. Not only this, US immigration check is so strict that when you leaves a country with a child (specially to Mexico), they check for proofs that the child belongs to you or not. Even if there is any mismatch in the last name of mother and child, you are in trouble. I got this information from internet.

So the bottom line is, I wanted my son's documents to not have FNU in mother's name.
We requested the same to the hospital staff but they said they can't leave the last name empty in the form. Also, they wanted us to show some document that shows my name as XYZ instead of FNU XYZ. Hospital records and insurance had my name as FNU XYZ.
But luckily my driver's license has my name as XYZ and last name is blank. Check out the DL story without last name.
So, finally hospital was ready to go with my name without last name.