Many of us have had the neighbor's from hell. Sure, you could move and per your CC&R's, you may have to, as a result. Fact is, you may not like this, but if you are not allowed to run a business, you very well may have to move OR give it up. You may know others that have the same business, but they may not have the same problem (bad neighbor) as you.
Try to put yourself in your neighbor's shoes. They may not like kids. Okay, fair enough. They have to listen to your kids make racket all day. I love kids, but hey, they can get noisy.
My kid threw a rock over the neighbor's fence the other day and she was none to happy about it. She said "Don't let them do it again!" So, after reading your comment, it seems it started with a similar incident.
You could go over and just talk to them. Be nice. Be pleasant. Take a batch of (store bought pastries or something...just to show you are neighborly!) It's ONLY a thought. Anyway, talk to them and ask them how you can work together to best resolve this situation. Be willing to listen.
You have kids that have obviously thrown items over the fence 2x. What if someone was hit...a pet, a person?? Again, put yourself in their shoes. Ensure their is nothing that can be thrown over the fence. Perhaps schedule a morning walk to the park, if they seem to complain about noise the same time in the day. Remind them that kids do enjoy playing and they have high energy...you can't control that! However, is there a time they are more sensitive to the children playing outside?
But, the point is, ask them what they would like to do, to work with you...that you want to be a good neighbor. You take care of kids in the daytime and it allows you to be a stay at home mom. They may understand that.
That's my suggestion.
If it doesn't work out...well, I would consider ignoring it otherwise. Seriously. (Allow them no more rent space in your head!) You are fully compliant - legally. Your PM knows you have the biz. Sounds like the only issue now is the HOA CC&R's and that could be an issue. Fact is, it's easy to find out. Call them up and ask. Keep the noise down AND be respectful of your neighbor's low noise tolerance. Simple! Plan a walk to the park...kids gets exercise and they can run, jump, scream, etc! (If there is an HOA, chances are there may be a park in your community.) Oh, and lastly, don't give your neighbor's any more ammo like rocks being thrown over the fence.
Lastly, if those neighbors are thorough, and it sounds like they are...make sure you are fully compliant in all matters, including taxes and whatever else is required...