Ozai seemed very prepared for the occasion. Also, if he would readily kill his own son, killing his own nephew shouldn't be that hard. Also, Ozai presumably green-lighted the plan to kill off an entire battalion of new solders, so he probably doesn’t care who is killed in battle. Maybe he had Zhao kill Lu Ten. Zhao, being as close to Ozai as he was, must have known of Lu Ten and Zuko's close friendship. He could have done it only to spite Zuko.