Parenting Question

If there was a neighborhood kid that your child was friends with, but at times this child was mean or manipulative to your child, but your child still wanted to play with this kid, would you allow him/her too?
At times this child seems to be Matthew's best friend, but then his mood will switch about what seems to be nothing and he's not his friend anymore, until he gets bored and wants to play again. I've explained to Matthew that he needs to ask for an apology and tell this kid to either be his friend all the time or not at all, but Matthew says this kid gets mad easily and it won't solve anything. He also wants to keep a friendship with this kid as they are neighbors. I don't feel like talking with his parent would be good, but it's getting really exhausting. I really care about the neighborhood kids, especially this one, but I'm really at my whits end with the drama this is bringing and the constant rejection my son is receiving. I know that my son is not perfect and am sure at times he may be the one that causes the problems, but I think on majority it seems this kid will just be having a bad day and letting it out on my kid.
What would you do?