Establishiing Paternity when the father of child is deceased

I am a Native American, age 26. When I was 16 years old, I gave birth to my son. We lived in Arizona at the time. He is now 10 years old. His father and I were not a couple at the time of his birth. He denied him and did not sign the birth certificate.
When my son was 8 months old, his father died in an automobile accident.
I am now trying to get Survivors Benefits from SSA for my son.
What I need to know is how do I go about getting DNA from
a person who is deceased?
My son's father's mother and father are both deceased. He is also buried on a Piute Indian Reservation.I have been struggling with this problem for so long, I just need advice and guidence with this situation.

After a decade, with no possibility of a grandparentage DNA test, it is unlikely that there is any way to obtain DNA for testing other than through an exhumation.