You've obviously been given a lot of advice here, and I can appreciate that you are currently in a stressful situation at home.
Before I start on any opinions of my own, I will explain our situation so you know where my opinions are coming from.
We have a Cocker Spaniel who is 8.5yrs... I also have a 5yr old and a nearly 4yr old with special needs. We have recently got a kitten.
Our Cocker was very much a Cocker who didn't like being handled - she hated being picked up, and still is not a dog who enoys cuddles/physical contact... she also doesn't trust children at all, due to some unfortunate experiences in her adolescence.
As my children grew from baby into toddlers and now young children, they were constantly reinforced not to approach or cuddle/stroke the dog without us being there with them. I do appreciate it was easier, as Honey was not a Puppy getting in to everything, but a mature dog who knew where her safe spaces were, and trusted us to manage any uncomfortable situations. We had (and still do have) some close calls - yes, a toddler fell on Honey's bed when she was in it, and yes, she growled and snapped - but as a pup we had worked very hard on bite inhibition and, for all her grumpiness (she is a very highly strung dog, and prone to guarding), she has excellent self-control. Both children have an excellent appreciation of respecting Honey's space and as we have always respected her growls (as in, seen them as a good warning that something is amiss and taught Honey ways to get out of the situation), Honey trusts us to keep her safe. It's hard work, but manageable.
Having had this experience, I can totally understand why many breeders do not like to home puppies with young children.
As far as advice goes - I would echo the advice of others to get a behaviourist who will come out and see you pup before making any judgements, and who uses only positive methods (and doesn't start spouting dominance). They can help you organise your house in a way that will give Betty the opportunity to relax away from the children etc, and assess whether the biting/growling is play, anxiety or guarding (and advice you appropriately on how to manage it). I would also stop *all* of you picking her up - if she needs to be moved, lure her away using a treat.... many pups of this age develop a dislike to being physically moved (understandably) and it is much better to use non-confrontational approaches to reinforce boundaries. Also, it will be easier to reinforce to your children not to pick her up if they don't see others doing it...
I wouldn't let unfamiliar children pet her.... if you are anxious, she will pick up on this and be *more* likely to react to a child as she will sense you are worried and think it is the approach of the child that is worrying you... this is how reactive behaviour can spiral. She has illustrated that she's not always confident about small children - if you take appropriate steps to ensure that she feels safe, that means she doesn't have to react (the only way she has to tell you she is uncomfortable is through growling or nipping, especially if she is on a lead as she would be in the shops).
Our kitten is now 5 months old and I naively assumed that 2 children used to respecting the space of a dog would understand not to harrass a cat.... but this was not correct, and I had to go back to basics with both children (but especially the youngest) about not picking her up and allowing her to rest etc... But we have got there... Yes, the children have been scratched - but this does not mean our kitten is aggressive (and neither does it mean that the children are badly managed!!!).
Managing a dog with young children can be hard work.
Managing a puppy can be hard work.
It *is* manageable, and nothing you have said sparks any major concern for me about Betty's temperament... she's a puppy in a busy situation. But, unless she is listened to and steps taken to stop putting her in situations she doesn't feel safe in, these things really could develop into a longer term behaviour problem. In short, you (and everyone around her) need to ensure Betty isn't put in positions where she may feel unsafe and respond by growling/biting, and you can then gradually desensitise her to those situations so that they become less stressful for her. However, she is still a puppy and will still bite/growl in play for a while to come - this is an important developmental stage for puppies and not something that can safely be cut short, as this is how they learn bite inhibition - the behaviourist should be able to explain some games/training exercises that can help reinforce this, though...
Good luck with whatever you decide to do.
