Balance in all things. Yes, make all interactions with the kids as positive as possible and controlled. But that doesn't mean you shouldn't correct him for making bad choices. The level of correction has to impress him without causing him fear or pain - when you cause fear or pain you loose trust.
Often behavior like this is caused by lack of socialization, lack of training and/or fear and insecurity. He is insecure about kids so he lashes out to keep them away. Every time he succeeds he learns being aggressive works so he does it more.
Though his attack on the child is extremely concerning and a huge red flag it doesn't mean that he is a lost cause. It just means more work on your part. He needs very clear boundaries, structure and rules. Training is the key. Find another trainer. I know plenty of trainers who are supposedly highly regarded but I wouldn't ever recommend their services. You need to find the right match for you, your family and the dog.
He should be on a leash attached to an adult as much as possible - in the house as well as outside. This gives you a mean of communication, correction, direction and an emergency brake.
This dog needs to learn to look to you for the answers when he has a decision to make. So instead of independently reacting by lunging and biting, he needs to look to you for guidance and you need to be ready with an answer. This means being highly aware of his energy, his body language and the surrounding environment. He needs to be excellent in his obedience skills so that you can be effective when you tell him to: sit, stay, down, come, stop, leave it, go to his place, etc.
Please feel free to contact us for help at any time. Talking can be more effective than posting because we can get into details more easily.
Love Them & Lead Them,
~Elizabeth & Doug
Dog Training the Way Nature Intended