Puppies are bloody hard work!
I got my oldest dog when he was a year old and even then he was a total and utter nightmare. Taking him to puppy training was really the best thing we ever did. He's now a very sweet, plodding 11 year old and the love of my life, but it took a lot of work (and money) to help him become a socially acceptable creature and even now he has his monster moments. I had many moments where I say and cried and wondered why we'd done this to ourselves but the work was worth it eventually.
OP, I can see where others have been frustrated as there are so many stories about poorly bred and poorly trained dogs out there. My two dogs have had really quite shitty backgrounds which is heartbreaking as they're really lovely dogs once the behavioural issues were dealt with.
BUT you're not the first person to have been manipulated into taking a dog from a horrible place. The people who sold you this dog played right into your hands, making sure that you were emotionally invested by seeing what a shithole the dog was in. That's why awareness of dodgy breeders and puppy farms should be spread everywhere.
I think perhaps you may have been a little naïeve in your choice of dog and especially in the timing, given that your little one was so young however, we all do make mistakes and we only ever realise its a mistake when it's too late. They do say that hindsight is 20:20 - it just doesn't help us at the time.
You've obviously hit crisis poin today and this really is where you're balancing on a knife edge. Some tough decisions need to be made.
Firstly, you really do need to clam yourself a bit and consider carefully whether you're keeping this dog or not. Make that decision based on the best thing for all of you (dog included) and be brutally honest with yourself about whether you can cope with this dog or not.
And please do make that decision with the proviso that if you do keep him, you both will put in 100% time, effort and money into this dog. If you decide to rehome, do so with a reputable no-kill rescue centre.
From there, if you decide to keep him then find a decent trainer or behaviourist who comes with recommendations and work your arse off to train this boy.
You've tried to train him yourselves but as a team (you, dp and dog) you've not got it right yet. There is time to get it right, but you will need help.
If you decide to rehome, centres may not be able to take him immediately and you may have to wait for a space to become available in a foster home or kennels for him. In this situation, it's important for you to separate dc and dog as much as possible.
It sounds like the dog is doing normal mouthing puppy behaviour but unfortunately for your dc, it's just at his face height and that's not good at all. Saying No and eyeballing the dog isn't going to be enough to break this behaviour and you'll need expert help with this.
While waiting for rehoming you'll still need to meet the dogs needs, so lots of exercise, stimulation and a good diet. Neither training nor rehoming will be simple, easy or quick so do think carefully about your next steps but whatever you decide, do it for the best interests of the dog and your dc. If you can't commit to outside help and time and money in training (and ongoing frustrations of owning a pup) then maybe it is time to admit defeat and give the dog up for rehoming.
Dogs are a major commitment and if you, despite your best intentions, can't give that full commitment, then the dog isn't for you. Please think carefully about your next steps and do get some advice from local and national rescues such as dogs trust if you can. You sound like you need help but you're going to have to seek it out (and probably pay for it)