I had this issue with Bear. When I first got him at 10 months he was ridiculous with the kids if they ran, crawled, jumped, wrestled, cried, etc. He was definitely the worst when the kids wrestled, fought, cried or a child was picked up. He would bark and mouth/nip at them. Sometimes it was excitement, sometimes it was anxiousness. Anyway, it sounds like your boy is in the same boat. Here are some things that we did, maybe it will help you.
For times I could not control the situation:
Bear was on leash and tethered to me (belt around my waist) whenever the kids may get loud. I knew certain times of the day that were most likely to cause an issue and would keep Bear tethered during those times. A few times a day we had quiet time that I knew the kids would be relaxed (watching a movie, reading books, etc) and those times Bear was off leash. When Bear was not leashed I basically had a leash attached to me ready to leash Bear if a fight broke out or someone started crying. It is not easy to ignore your children if they are full out brawling or crying but that is what I had to do until I could get bear restrained and in his crate or behind the gate. I had gates placed around the downstairs (the hotbed of activity) so I could quickly put Bear on the other side of a gate so he could still observe the commotion but not be in the middle of it. Space is your friend so when the kids do something that triggers him, create enough space that he is not reacting. The goal here is to keep the dog from reinforcing the bad behavior of biting when the kids do one of his triggers. If I was too busy to keep a hawk eye on Bear, he was separated from the kids.
For times I could control the situation:
Probably 2 times a day we worked on desensitizing Bear to the triggers. I knew picking up children, kids fighting and kids crying were 3 of his top triggers. I would fill my treat bag of really good treats and have the kids act out the behavior and we worked on one thing at a time. We started with fighting and I had the kids act out fighting with each other while I had Bear on leash 2 rooms away. I had to find the distance that he could watch them but not react (pull towards them). I had the kids start fighting and I gave him treats over and over. I would tell the kids to stop and all treats would stop. Then I would tell them to start again and I would feed him the treats. Eventually he would look to them and then look back at me for the treats. I would also occasionally do obedience training with him while the mehem was occuring in the other room so I knew I could get his focus over the kids. I did this over and over slowly moving towards the kids until we were literally 1 foot from the kids wrestling on the floor. We stayed that close for awhile before I allowed him off leash when the kids were play fighting. Most of the time I put him in a down so I knew he could relax better. It probably took 2 weeks but now my kids can full out brawl right next to him and he would just lay there. I retrained him that kids fighting is a good thing and I occasionally still toss him a treat when the kids are all crazy to reinforce it (always have treats around the house). We did this with every one of his triggers. Picking up kids probably took the longest one to break but now when I pick up my son he goes to bark and then remembers if he sits quietly he gets something yummy. You can see him thinking about it. He overrides his previous behavior and his but hits the floor and he looks to me for his treat. Kid picked up = sit for yummy treat. Eventually after some time of intermittent treats you can cut the treats off because it becomes muscle memory. Kid picked up = sit.
Once we retrained Bear with all our triggers we had to train him that the same goes for other kids as well but that was a very quick process. In all it took us a few weeks of consistent work to desensitize him to those 3 situations. Bear actually had a lot more triggers (crawling, jumping, riding bikes, sledding, swinging) but those 3 were the worst. We trained him the same way on every one of his triggers and now he is a totally different dog. It absolutely can be done. You just have to train the dog a new behavior when the triggers are presented.
Another thing I did with Bear is we have a no mouthing rule with everyone. NO ONE is allowed to play rough with Bear and he is NOT allowed to put his mouth on anyone. As soon as he tried to mouth someone he was put in a time out. I didn't say a word to him, just calmly removed him from the room for a couple minute time out and repeated if he tried to mouth again. You need to have no tolerance for mouthing even in play with adults.
I do want to say that I do not believe in harsh corrections for this type of behavior. I seriously contemplated using a shock collar or water spray bottle to deal with this before I read the consequences of such corrections. One thing I knew was that I did not want Bear to associate pain/negativity with my children. That is why I believe teaching him a different behavior and desensitizing is the right way to go here. You want the children to equal positive things.
