Guess I'm old school, talk and tender footin around is to PC for me. I'd start walking towards him, telling him he has 2 choices turn the kid loose or die. Because I'm going to kill him unless he gets the kid turned loose first. And I would do IT! And I think it will show. And I will be close enough to make the shot when I take it, because the barrel may be touching his eye when the gun goes bang.
Maybe I have watched to many Dirty Harry movies. But stale mates don't go well most times. So you may as well screw with his OODA loop and throw it for a tail spin before he has time to think. His thoughts are you will pee in your pants and beg for mercy, when you don't, you have about 2 seconds to end it.
I know some if not most here will say I'm wrong but sometimes you have to let the apples fall as they will.
Can I live with the loss of my child? I think I can because I tried but can you live with the loss if you do nothing and he still kill your child. Because what I see is if you talk and talk and talk at some point he is going to tell you to put down your gun or he will kill the child. Question do you do it or let him shoot you child? If you do you all may or will die. You have already shown you don't want to act but talk and comply, he has control of the situation. You've lost before you got started.
Now the big question here is do YOU have the mindset to pull this action off? I would guess most here do not. Why? Because "It's gotta be who you are, not a hobby".