I couldn't find Obsorge in any of my legal dictionaries (Dietl, Romain), but it does mean Pflege, sorgende Aufsicht which is custody or care (as in elterliche Sorge = parental custody). So you're fine on that count.
However, it is not current custody but the request of *each* or both the parents for sole custody that is being rejected, meaning that the court probably feels that it is in the best interests of the child not to award sole custody to any one parent.
The definition of Pflegschaft in Dietl is: Pflegschaft is the care entrusted to a curator appointed by the guardianship court to look after particular interests (particularly property interests) of another. The equivalent it gives is curatorship (caretakership would be better here) or trust. However in your case it does seem easiest to refer to it as a custody case unless there are any property aspects involved.
Kindschaftsrecht is (acc. to Dietl) the law of parents and children. That sounds terribly awkward, but I think it goes beyond merely custody legislation.
You could try Child Welfare Law, Family Law or The Law Of Parent-Child Relationships (website references below). Your sentence could be: "The judgement is based on Child Welfare Law in the United States entailing that the child's best interests be taken into account while arriving at a decision" or something like that.