1.5 year old rescue adoptee is nipping at 3 y/o unprovoked - what to do?

I feel terrible about this. After months of considering it we finally went ahead and adopted loveable lab mix from the local Humane Society. He is unimaginably sweet with me and my husband, very affectionate and cuddle and so loyal.

However, with my 3 year old, he nips at her. He has yet to actually bite her, but i'm worried that it is in the future.

Originally I thought she was doing something to him to encourage his behaviour, since she's 3, but the last couple times she was with me and petting him int he same manner as I was and he suddenly head turned and nipped at her. He nipped enough to take the nail polish of her nail, but not enough to leave a mark.

It has gotten that my 3 year old is increasingly apprehensive of him and will only pet him from an extended far away distance and only when I am.

We got the dog to add to our family and provide a companion and friend for our daughter and I'm at a loss as to what to do.

I've contacted to human society for advice, we've only had him for 2 weeks. 1 week of fostering and 1 week since adoption was finalized.

Any advice or guidance is appreciated. This little guy has been abandoned once already, we love him, but want him to be a safe and happy addition to our family.

First and foremost, a match for you and hubby might not be a match for your 3 yr old.
Having said that, you have lots of work to do. This dog might not have been raised with children and has shown that he is willing to go to 'teeth' to keep her at bay. Often dogs who are not raised with kids think they are little aliens and make the dog very nervous. Children move erratically, fall unpredictably, make loud noises, have lots of energy, and look different from adults. It might well be that your child petted too intensely, or squealed a bit, or moved too quickly or got too close. Your dog did what made him feel safe in the moment. NOT excusing it, just trying to understand it.
Never leave your child alone with this dog - do not even turn your back. Only because 3 yr old kids are impulsive and could make a mistake, or the dog could simply make a mistake without much provocation.
This is an opportunity to teach your child how to behave with a dog, NO kisses, NO grabbing, NO Squealing, NO jumping into the dogs space, NO taking toys/bones from dogs, NO pulling on hair, NO hugging. Only calm energy, and lots of space between them unless you are present to take charge.
Find a great trainer to help you teach your dog boundaries, skills and manners. If your dog feels that you are keeping the child at bay then he won't feel the need to step up. If he can do a long down stay in the family room while you all play in the room then he can be part of the family without being in the center of the commotion - which can help him desensitize to the energy of a child. A well trained 'leave it' can help him to learn to back away before he even thinks about getting nervous.
Most importantly, if you step up and teach this dog how to behave then he will begin to look to you for advise when he is insecure/nervous, and if you are ready with the answer/guidance then things can smooth out quickly. The more a dog understands the more confident and calm he should become.