This man may have a very good case. The child's immigration status is a dependant of the father, which is a HUGE advantage for the man. What is the woman's current imigration status? The income of each party before divorce was not relevant. The current higher income of the man may not be an advantage at all.
usabc2991 was wrong. Practically in US, women always gain sympathy in divorce cases. The male party must have a solid superiority to gain sole custody of the children.