ADDED: I'm just amazed that some people are posting that this woman is an "idiot," that you should childishly tell her she has something in her teeth or whatever (immature beyond belief), or worst of all, you should tell your son "SHE has problems" and so on. They're simply taking every single word of a fourth grader at face value. P., I hope you don't listen to those kinds of posts and choose instead to think before you make assumptions or lash out as they advise. Listen to posters like Terri S. and Suz T.
Original post: First, congratulations on the hard work you and your son have put in to help him improve at school and move forward. I think all of us here can acknowledge that.
And now, please try to step back from your own strong, defensive emotions and acknowledge for yourself that you are feeling sensitive and defending your son against what you see as an insult, when you have no objective knowledge that any insult took place. You wrote your post when you were upset and venting hard, which is understandable.
But can you be a little more objective now and see that outsiders like us here would say that maybe you are taking the word of a fourth grader too seriously? I am not saying your son is lying. I am saying he's still young and still not experienced in questioning what others say or navigating social situations. So when he says the other boy told him "I can't come over because you have problems" or whatever -- the boy might have misheard something at home; the boy might not want to play with your son and made that excuse up because it prevents him from having to say "I just don't want to play with you"; the boy might not have said quite that at all but your son heard it and sincerely believes it's what was said.
It's telling that you mention earlier that your son told you the other boy was allergic to cats, so you hadn't invited the boy over due to that. Did you think to call the other mom and ask? Your son could have just had it wrong and the other boy might not be allergic at all. I'm not saying "don't trust your son," I'm saying -- kids get things wrong and parents need to check up on things. That includes this "you've got a problem" thing, now. It sounds like you take your son's statements about other people at face value. But you're the adult and more experienced, and it's up to you to help him navigate interactions for a while to come.
Can you see how, when it came to planning this play date that did not happen, you left it all up to others, especially your son? He told you the other mom would call. When she didn't, you kept asking him about it for day after day, rather than contacting her yourself. This other mom might just have flaked out and forgotten, but if your son wanted the play date -- why didn't you call? Maybe you were concerned about being seen as pushing too hard, but it isn't like you had been pursuing her hard for a play date already so that shouldn't be a worry.
So invite the other boy. With a very cheerful voice when you call (and I would call and not e-mail or text. Using e-mail or text frankly makes it easier for the other mom to say no; in person is best, or by phone). She may surprise you by saying yes. If she says no, don't press too hard, but I would probably say, with utter politeness because you do not know what she might have said if anything at all: "Son mentioned that Your Son seemed reluctant to have a play date with him. Is there something going on between the boys that we can help them with, or can I answer anything for you?"
In short: Give her the benefit of the doubt. Assume the best, not the worst, especially as you don't really know her personally at all, it seems. She may indeed be a witch, but you don't know that first-hand -- you think you heard what a fourth grader thinks he heard from a friend who maybe thinks he heard something from his mom. Even if the woman is indeed awful -- you at least will have taken the high road, and modeled restraint for your son, by not going ballistic on her.