I know it's a big ask, and without knowing the child I can't say whether it it's a good idea or not. Just a suggestion.
I think it might be good idea for zippyb to have a word with the teacher first before sending a note if she decided to do this. I just think the mother ( assuming she cares about such things) could feel awful in either case and at least by sending a friendly note, expressing the hope that the children could meet for tea, you are opening lines of communication with the mother and not being totally hostile. If you do nothing and the mother finds out, she is going to think the worst, how could she not?
And if her son picks on the zippyb's boy, knowing he was the only one not to get an invite to the party could make things even worse. At least an explanation and hope for a future get together might help. I'm not saying a note will make everything right, but at least it could lessen hostility. Or failing that, give a factual reason - 'you hit my son' - for the lack of invite.
I have a friend and her son is best friends with my son and they play well together reguarly, but he rarely get invites to his birthday parties, although I invite him to my sons. It's a matter of logistics (she lives an hours drive away) and she likes to invite just his group of school friends, and my son goes to a different school. I would have been hurt if she had said nothing and also annoyed at her for being underhand, since they are such good friends, but the mother told me her reasons, they make practical sense and that's OK. Instead we arrange for the boys to meet up for an afternoon later.
Also the note can convey the feeling that the mother is not being blamed for her son's actions. It is just a simple matter that at the moment the 2 boys don't get on very well and the wish to avoid a bad scene. No note and the mother is (well I would) might feel harshly judged as well.