You should have said, "Okay coach, you win, it was a double dribble"
Then he would have looked at you very confused.
Oh and to answer your question, it's a double dribble. As BNR was walking you through how it could have been a travel first, had he lifted his pivot prior to the 2nd dribble, then it would have been a travel. Nothing further needed to happen on the play, violation, move on. But as his feet were unmoved, it becomes a double dribble when he is the first to touch.