Sorry about that, I only noticed your question today. That Social Security doc you want is up on Google images.
https://wtpotus.files.wordpress.com/2013/08/sad1-copy.jpg
it doesn’t look like any big deal, all she had to do was supply the name of her previous registration.
OK, here’s the answer to whether she had to get a court order or not:
She filled out the card January 3, 1963, and her birth date was Nov 29, 1942.
At the time of filling it out, she was only 20 years old.
So that meant that she didn’t need a court order to do the initial name change from Stanley Ann Dunham to Ann Dunham Obama; she would only have needed to go down to the county recorder with her parents, since she was under 21.
If she had been over 21, she would have needed a court order.
That simplifies things.