You have gotten a lot of great advice. I would not call the other mother, don't extend an invitation to her son. If his mother does allow him over to play again tell him he can't play until he tells the truth about how he threw a ball at your son, but he needs to tell the truth in front of you, your son, and his parents. If he is not willing to do this then he is not welcome in your yard.
However, if he does tell the truth, lay down the ground rules in front of his parents so all involved know what is expected and let him know that if he lies again he will be banned from your yard until he fesses up.
You might also think about asking him to come to you if your kids say anything that upsets or bothers him. You may not have heard or seen something that your kids said that set this boy off. Sometimes the statement can seem innocent, but hits a nerve. You wrote that the boy was angry because the other kids wouldn't play the game he wanted to play, but how did your kids say they didn't want to play? Sometimes the way it was said can trigger an attack.
My son, who is 6, is very sensitive and when I question him about an incident it usually comes to light that he said something rude to the other boy that caused an altercation. The other boy should not have gotten physical, but my son shouldn't have gotten mouthy either.
I like the idea of a hidden video camera, but I don't know if you could pull it off without your kids finding it. If you can, good luck, I'd use it, but I don't know about the legality of it.
Overall it can be hard to deal with other people's kids, especially when the parents won't back you up. We have had a few problems with our neighbor's grandson, but we have just sent him home when he misbehaves. The last time he wanted to come over and play his father said "No, you always get into an argument when you go over to their place." Which is true, he doesn't like the rules and doesn't like it when we stick to them.
If he can behave he is welcome to come over, but until then we will continue to send him home.
Good luck with your neighbors.