OP
I am not saying this person is necessarily a good friend to you, I don't really know. It sounds like she doesn't always do the right thing by you, but has a good side too and has been caring towards you in the past. Only you can know whether the good outweighs the bad, and so can decide if her friendship is important to you (or important enough to try to find a way past this incident).
I am also not saying her DD's behaviour was ok and that it should not have been challenged- quite the opposite. You, or DH, should have either talked to the school about it or spoken to your friend directly.
But I really think that your DH having a "quiet word" with the child of a woman who is known to find him intimidating, and whom he despises to the point he cannot usually even bring himself to acknowledge her presence, was never going to go well. I also suspect that your DH knows she finds him intimidating, which makes his actions even more difficult to put a benign slant on, TBH (unless you are absolutely sure he does not know she finds him intimidating). I genuinely don't think this could ever have worked out as a "friendly chat" given their history and I am surprised that he would have thought that it would be.
You also only have his word that the chat went as he says it did. Being objective- he may have relayed the "chat" he had with your friend and her DD to you exactly as it happened and you ex-friend has over-reacted (because she has a tendency to do this, or perhaps because of the difficult relationship between her and your DH, or perhaps both). It is also possible that he relayed it accurately from his point of view, but that he is not aware that he can come across as intimidating/stand offish and so he didn't realise he had made your friend feel intimidated or upset (for example, when he says she agreed with him as a demonstration that all went well that may not be reliable- sometimes people do this when they are upset/frightened, just to try and get things over and done with). Alternatively, your DH doesn't like this woman, is upset that her DD has been unkind to your DS and also lied to you about it and so chooses to "have a quiet chat" with her DD in the full knowledge that this woman finds him intimidating and uses that to his advantage. Or perhaps he intends to have a quiet word, but given the animosity towards her it becomes a little more forceful or angry than he intended/realised. I have no idea which of those scenario's is the case- you know them well, so perhaps you are best placed to know
Look, I imagine anyone challenging a child and basically telling them off would result in the child crying, so the fact the child cried is a bit of a red herring. That said I personally wouldn't approach a child in this manner and tell them off because it doesn't seem a sensible approach (what if your DS hasn't quite told you the whole truth, for instance. I'm not saying he has in this instance but as a general point). I'd speak to the parents or the school, as I said.
I also think that if your ex-friend was genuinely was feeling intimidated or upset about what happened, why should she not speak to the school about it? Why does she need to speak to you about your DH's behaviour towards her first- what could you say that would make that ok? Do you think that you saying she was upset about the incident change his behaviour? Or is it that you actually think she complained to the school to be malicious because she doesn't like your husband?
Overall, I do think your DH was unwise to tackle this directly. I also think you need to try and get a wider group of friends (and this may mean getting some help with your social anxiety, if you think that is what is holding you back). I think it sounds like your friendship with this woman has run it course, and there's probably been errors on all sides TBH.
BUT I'd also say your DH does sound like he doesn't; always treat your well and can be a bit of a bully. Do you think that's a fair assessment? I think your DH also needs to work on how he comes across to others if it really is the case that other parents, who you'd like to be friendly with, keep their distance from you when your DH is around. For quite a few people to do that- to the extent that it's not really that they just say a quick hello and move on, but that they barely acknowledge you or act like they don't know who you are (especially if they usually say hello/have a chat), then my conclusion would be that he makes others really very uncomfortable. That's something of a concern to me and I think you do need to address that with your DH. Is he aware of how he comes across? If so, does he care?