Why Did The AP Publish A Dying Soldier's Photo?

Because they are assholes.

“I cannot imagine the pain and suffering Lance Corporal Bernard’s death has caused his family. Why your organization would purposefully defy the family’s wishes knowing full well that it will lead to yet more anguish is beyond me. Your lack of compassion and common sense in choosing to put this image of their maimed and stricken child on the front page of multiple American newspapers is appalling. The issue here is not law, policy or constitutional right – but judgment and common decency.”