Well for one I think you are correct. Your dog is in puppy mode still.
He is fighting for dominence with her. OR she was pestering him. Either way you must teach the child to A. Not tease him. and B. Never put her face near his face.
I have a Jack Russell Terrier. A very loving playful one but aggresive, dominate, and queen of the house. Jack Russells, like Chow Chows aren't childrens dogs. They do NOT tolerate teasing, loud screeching from kids, rough play with kids etc. So when my roomate started bringing his 5 year old son over you can see where I was concerned.
At first she nipped him when he would get excited. Calvin would yell "daddy" in a high tone and she would bite him. Eventually after punishments and making her obey Calvin it stopped.
Calvin gives Sonique her food when he is here. She depends on him. Calvin tells her to sit, stay, not to beg, etc when he is here. She knows she must listen to him. This has really helped. And keep taking him out in public, LOTS. And to parks with kids yelling (protective muzzle or leash of course) and lots of places where he will get used to anything.
In the end of course YOU must decide how much danger this dog presents to you and your family. But when you get a dog, you agree to take ALL his problems. And if this means you maybe have to pay for classes for him, and your daughter to attend together, or something else. Then do it. Don't let another dog loose and home, or a child her nose!
I am glad you are willing to work with him. Try asking questions on dog newsgroups, dog forums, message boards etc. There is a lot of great free advice out there.
Good luck! I know what its like having an aggresive dog!